// Numbas version: exam_results_page_options
{"name": "Integration revision", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "percentPass": 0, "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Basic", "pickQuestions": 1, "questions": [{"name": "Basic integration 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "rulesets": {}, "metadata": {"description": "", "licence": "None specified"}, "ungrouped_variables": ["a", "b", "c", "a1", "d", "a2", "a3", "a4", "a5"], "tags": [], "statement": "<p>Integrate each of the following functions with the given limits.</p>\n<p>Give your answers correct to at least 3 significant figures.</p>", "parts": [{"prompt": "<p>Integrate and evaluate</p>\n<p>\\[\\int_\\var{c}^\\var{c+2}\\simplify{{a}x-sin({b}x)}dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a1}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "<p>First integrate indefinitely</p>\n<p>\\[\\int\\simplify{{a}x-sin({b}x)}dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a}*x^2/2+1/{b}*cos({b}*x)", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "<p>Integrate and evaluate</p>\n<p>\\[\\int_\\var{a}^\\var{a+2}\\simplify{{b}/x^{c+2}+{d}*sqrt(x)}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a2}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "<p>First integrate indefinitely</p>\n<p>\\[\\int\\simplify{{b}/x^{c+2}+{d}*sqrt(x)}dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "-{b}/{c+1}*x^{-c-1}+{2*d}/3*x^(3/2)", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "<p>Integrate and evaluate</p>\n<p>\\[\\int_0^\\var{b}\\simplify{{a}*exp(x/{c+1})+{d}}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a3}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "<p>First integrate indefinitely</p>\n<p>\\[\\int\\simplify{{a}*exp(x/{c+1})+{d}}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a*(c+1)}*exp(x/{c+1})+{d}*x", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "<p>Integrate and evaluate</p>\n<p>\\[\\int_\\var{c}^\\var{c+1}\\simplify{{d}/({a}*x)+{a}/{c}*cos({d}*x)}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a4}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "<p>First integrate indefinitely</p>\n<p>\\[\\int\\simplify{{d}/({a}*x)+{a}/{c}*cos({d}*x)}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{d}/{a}*ln(x)+{a}/{d*c}*sin({d}*x)", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "<p>Integrate and evaluate</p>\n<p>\\[\\int_0^\\var{c}\\simplify{x^{a}/{b}+{c}*exp(-{b}*x)-{d}}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a5}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "<p>First integrate indefinitely</p>\n<p>\\[\\int\\simplify{x^{a}/{b}+{c}*exp(-{b}*x)-{d}}\\;dx\\]</p>", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "x^{a+1}/{b*(a+1)}-{c}/{b}*exp({-b}*x)-{d}*x", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "", "variables": {"d": {"name": "d", "description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables"}, "a2": {"name": "a2", "description": "<p>Answer to question 2.</p>", "templateType": "anything", "definition": "(-b/(c+1)*(a+2)^(-c-1)+2*d/3*(a+2)^1.5)-(-b/(c+1)*a^(-c-1)+2*d/3*a^1.5)", "group": "Ungrouped variables"}, "a1": {"name": "a1", "description": "<p>Answer to question 1.</p>", "templateType": "anything", "definition": "(a/2*(c+2)^2+1/b*cos(b*(c+2)))-(a/2*c^2+1/b*cos(b*c))", "group": "Ungrouped variables"}, "a4": {"name": "a4", "description": "<p>Answer to question 4.</p>", "templateType": "anything", "definition": "(d/a*ln(c+1)+a/(d*c)*sin(d*(c+1)))-(d/a*ln(c)+a/(d*c)*sin(d*c))", "group": "Ungrouped variables"}, "a5": {"name": "a5", "description": "<p>Answer to question 5.</p>", "templateType": "anything", "definition": "(c^(a+1)/(b*(a+1))-c/b*exp(-b*c)-d*c)-(-c/b)", "group": "Ungrouped variables"}, "b": {"name": "b", "description": "", "templateType": "anything", "definition": "random(1..9 except a)", "group": "Ungrouped variables"}, "a": {"name": "a", "description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables"}, "c": {"name": "c", "description": "", "templateType": "anything", "definition": "random(1..5)", "group": "Ungrouped variables"}, "a3": {"name": "a3", "description": "<p>Answer to question 3.</p>", "templateType": "anything", "definition": "(a*(c+1)*exp(b/(c+1))+d*b)-(a*(c+1))", "group": "Ungrouped variables"}}, "type": "question"}]}, {"pickingStrategy": "all-ordered", "name": "Substitution", "pickQuestions": 1, "questions": [{"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "n", "b", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>Identify the inside function. $u=$ [[0]]</p>\n<p>Differentiate: $\\frac{du}{dx}=$ [[1]]</p>\n<p>Make $dx$ the subject: $dx=$ [[2]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "x^({n}+1)+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "({n}+1)*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "du/(({n}+1)*x^{n})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<table style=\"float: left;\">\n<tbody>\n<tr>\n<td>Rewrite the whole integral in terms of $u$ and $du$: $\\int$ [[0]]</td>\n<td style=\"text-align: center; border-bottom: solid;\">$du$</td>\n</tr>\n<tr>\n<td></td>\n<td>[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"clear: left;\">Simplify and cancel $x$'s: $\\int$ [[2]] $du$</p>\n<p style=\"clear: left;\">Integrate with respect to $u$: [[3]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}*sin(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({n}+1)*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/({n}+1)*sin(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/({n}+1)*cos(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}*x^{n}*sin(x^({n}+1)+{b})}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/({n}+1)*cos(x^({n}+1)+{b})+c", "steps": [{"showCorrectAnswer": true, "prompt": "<p>Replace $u$ with $\\simplify{x^({n}+1)+{b}}$ in the previous step.</p>", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 1 decimal place:</p>\n<p>\\[\\int_0^\\pi\\simplify{{a}*x^{n}*sin(x^({n}+1)+{b})}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "1", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral can be evaluated by using&nbsp;<span style=\"text-decoration: underline;\">substitution</span>:</p>\n<p>\\[\\int\\simplify{{a}*x^{n}*sin(x^({n}+1)+{b})}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..8#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-5..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "-{a}/({n}+1)*cos(pi^({n}+1)+{b})+{a}/({n}+1)*cos({b})", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "n": {"definition": "random(1..4#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by substitution 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "n", "b", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>Identify the inside function. $u=$ [[0]]</p>\n<p>Differentiate: $\\frac{du}{dx}=$ [[1]]</p>\n<p>Make $dx$ the subject: $dx=$ [[2]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "x^({n}+1)+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "({n}+1)*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "1", "showCorrectAnswer": true, "scripts": {}, "answer": "du/(({n}+1)*x^{n})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<table style=\"float: left;\">\n<tbody>\n<tr>\n<td>Rewrite the whole integral in terms of $u$ and $du$: $\\int$</td>\n<td style=\"text-align: center; border-bottom: solid;\">[[0]]</td>\n<td>$\\times$</td>\n<td style=\"text-align: center; border-bottom: solid;\">$du$</td>\n</tr>\n<tr>\n<td></td>\n<td>[[4]]</td>\n<td></td>\n<td>[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"clear: left;\">Simplify and cancel $x$'s: $\\int$ [[2]] $du$</p>\n<p style=\"clear: left;\">Integrate with respect to $u$: [[3]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.25", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.25", "showCorrectAnswer": true, "scripts": {}, "answer": "({n}+1)*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/(({n}+1)*sqrt(u))", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "2*{a}/({n}+1)*sqrt(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "sqrt(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}*x^{n}/sqrt(x^({n}+1)+{b})}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": "0.001", "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "2*{a}/({n}+1)*sqrt(x^({n}+1)+{b})+c", "steps": [{"showCorrectAnswer": true, "prompt": "<p>Replace $u$ with $\\simplify{x^({n}+1)+{b}}$ in the previous step.</p>", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to the nearest whole number:</p>\n<p>\\[\\int_0^{10}\\simplify{{a}*x^{n}/sqrt(x^({n}+1)+{b})}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "0", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral can be evaluated by using&nbsp;<span style=\"text-decoration: underline;\">substitution</span>:</p>\n<p>\\[\\int\\simplify{{a}*x^{n}/sqrt(x^({n}+1)+{b})}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(1..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "2*{a}/({n}+1)*(sqrt(10^({n}+1)+{b})-sqrt({b}))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "n": {"definition": "random(1..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by substitution 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "n", "b", "ans", "m"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>Identify the inside function. $u=$ [[0]]</p>\n<p>Differentiate: $\\frac{du}{dx}=$ [[1]]</p>\n<p>Factorise this: $\\frac{du}{dx}=$ [[2]] $($[[3]]$)$</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "x^{m}+{m}*{b}*x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{m}*x^({m}-1)+{m}*{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{m}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "x^({m}-1)+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<table style=\"float: left;\">\n<tbody>\n<tr>\n<td>Rewrite the whole integral in terms of $u$ and $du$: $\\int$</td>\n<td>[[0]]</td>\n<td style=\"text-align: center; border-bottom: solid;\">$du$</td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>[[1]] $($[[2]]$)$</td>\n</tr>\n</tbody>\n</table>\n<p style=\"clear: left;\">Simplify and cancel $\\simplify{x^({m}-1)+{b}}$: $\\int$ [[3]] $du$</p>\n<p style=\"clear: left;\">Integrate with respect to $u$: [[4]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}(x^({m}-1)+{b})*u^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.25", "showCorrectAnswer": true, "scripts": {}, "answer": "{m}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.25", "showCorrectAnswer": true, "scripts": {}, "answer": "x^({m}-1)+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/{m}*u^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/({m}*({n}+1))*u^({n}+1)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}(x^({m}-1)+{b})(x^{m}+{m}*{b}*x)^{n}}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": "0.001", "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/({m}*({n}+1))*(x^{m}+{m}*{b}*x)^({n}+1)+c", "steps": [{"showCorrectAnswer": true, "prompt": "<p>Replace $u$ with $\\simplify{x^{m}+{m}*{b}*x}$ in the previous step.</p>", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 4 significant figures:</p>\n<p>\\[\\int_0^1\\simplify{{a}(x^({m}-1)+{b})(x^{m}+{m}*{b}*x)^{n}}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "4", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "sigfig", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral can be evaluated by using&nbsp;<span style=\"text-decoration: underline;\">substitution</span>:</p>\n<p>\\[\\int\\simplify{{a}(x^({m}-1)+{b})(x^{m}+{m}*{b}*x)^{n}}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "ans": {"definition": "{a}/({m}*({n}+1))*((1+{m}*{b})^({n}+1))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "b": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "m": {"definition": "random(2..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "m", "description": ""}, "n": {"definition": "random(3..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"pickingStrategy": "all-ordered", "name": "Parts", "pickQuestions": 1, "questions": [{"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>The product $\\simplify{{a}x*cos({b}*x)}$ must be written in the form $u\\frac{dv}{dv}$.</p>\n<p>Determine $u,\\frac{du}{dx},\\frac{dv}{dx}$ and $v$:</p>\n<table>\n<tbody>\n<tr>\n<td>$u=$</td>\n<td>[[0]]</td>\n<td>$\\frac{du}{dx}=$</td>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>$v=$</td>\n<td>[[3]]</td>\n<td>$\\frac{dv}{dx}=$</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "cos({b}x)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "sin({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Apply the integration by parts formula:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\int{u\\frac{dv}{dv}dx}=$</td>\n<td style=\"text-align: center;\">$uv$</td>\n<td>$-\\int$</td>\n<td style=\"text-align: center;\">$v\\frac{du}{dx}$</td>\n<td>$dx$</td>\n</tr>\n<tr>\n<td>$\\int{u\\frac{dv}{dv}dx}=$</td>\n<td>[[0]]</td>\n<td>$-\\int$</td>\n<td>[[1]]</td>\n<td>$dx$</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x*sin({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*sin({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}x*cos({b}*x)}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "musthave": {"message": "<p>As this is an indefinite integral, $+c$ is required.</p>", "showStrings": false, "strings": ["+c"], "partialCredit": 0}, "scripts": {}, "answer": "{a/b}*x*sin({b}*x)+{a/b^2}*cos({b}*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral:</p>\n<p>\\[\\int_0^1\\simplify{{a}x*cos({b}*x)}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "2", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "<p>Consider the following integral:</p>\n<p>\\[\\int\\simplify{{a}x*cos({b}*x)}\\,dx\\]</p>\n<p>This may be evaluated by using&nbsp;<span style=\"text-decoration: underline;\">integration by parts</span>.</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(1..3#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "{a}/{b}*sin({b})+{a}/{b}^2*(cos({b})-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts 2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>The product $\\simplify{{a}x*exp({b}*x)}$ must be written in the form $u\\frac{dv}{dv}$.</p>\n<p>Determine $u,\\frac{du}{dx},\\frac{dv}{dx}$ and $v$:</p>\n<table>\n<tbody>\n<tr>\n<td>$u=$</td>\n<td>[[0]]</td>\n<td>$\\frac{du}{dx}=$</td>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>$v=$</td>\n<td>[[3]]</td>\n<td>$\\frac{dv}{dx}=$</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "exp({b}x)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "exp({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Apply the integration by parts formula:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\int{u\\frac{dv}{dv}dx}=$</td>\n<td style=\"text-align: center;\">$uv$</td>\n<td>$-\\int$</td>\n<td style=\"text-align: center;\">$v\\frac{du}{dx}$</td>\n<td>$dx$</td>\n</tr>\n<tr>\n<td>$\\int{u\\frac{dv}{dv}dx}=$</td>\n<td>[[0]]</td>\n<td>$-\\int$</td>\n<td>[[1]]</td>\n<td>$dx$</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x*exp({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*exp({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}x*exp({b}*x)}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "musthave": {"message": "<p>As this is an indefinite integral, $+c$ is required.</p>", "showStrings": false, "strings": ["+c"], "partialCredit": 0}, "scripts": {}, "answer": "{a}/{b}*x*exp({b}*x)-{a}/{b}^2*exp({b}*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 2 decimal places:</p>\n<p>\\[\\int_{-1}^1\\simplify{{a}x*exp({b}*x)}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "2", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral may be evaluated by using&nbsp;<span style=\"text-decoration: underline;\">integration by parts</span>.</p>\n<p>\\[\\int\\simplify{{a}x*exp({b}*x)}\\,dx\\]</p>\n<p><strong>Note: $e^{\\simplify{{b}*x}}$&nbsp;</strong><strong>can be entered by typing exp($\\simplify{{b}*x}$).</strong></p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-8..8 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "{a}/{b}*(exp({b})+exp(-{b}))-{a}/{b}^2*(exp({b})-exp(-{b}))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts 3", "extensions": [], "custom_part_types": [], "resources": [["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "ans", "n"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>The product $\\simplify{{a}x^{n}*ln({b}*x)}$ must be written in the form $u\\frac{dv}{dv}$.</p>\n<p>Determine $u,\\frac{du}{dx},\\frac{dv}{dx}$ and $v$:</p>\n<table>\n<tbody>\n<tr>\n<td>$u=$</td>\n<td>[[0]]</td>\n<td>$\\frac{du}{dx}=$</td>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>$v=$</td>\n<td>[[3]]</td>\n<td>$\\frac{dv}{dx}=$</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "ln({b}*x)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "1/x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^({n}+1)/({n}+1)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Apply the integration by parts formula:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\int{u\\frac{dv}{dv}dx}=$</td>\n<td style=\"text-align: center;\">$uv$</td>\n<td>$-\\int$</td>\n<td style=\"text-align: center;\">$v\\frac{du}{dx}$</td>\n<td>$dx$</td>\n</tr>\n<tr>\n<td>$\\int{u\\frac{dv}{dv}dx}=$</td>\n<td>[[0]]</td>\n<td>$-\\int$</td>\n<td>[[1]]</td>\n<td>$dx$</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^({n}+1)*ln({b}x)/({n}+1)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}/({n}+1)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}x^{n}*ln({b}*x)}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "musthave": {"message": "<p>As this is an indefinite integral, $+c$ is required.</p>", "showStrings": false, "strings": ["+c"], "partialCredit": 0}, "scripts": {}, "answer": "{a}*x^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}x)-1)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 3 significant figures:</p>\n<p>\\[\\int_1^2\\simplify{{a}x^{n}*ln({b}*x)}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": 3, "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "sigfig", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral may be evaluated by using&nbsp;<span style=\"text-decoration: underline;\">integration by parts</span>.</p>\n<p>\\[\\int\\simplify{{a}x^{n}*ln({b}*x)}\\,dx\\]</p>\n<p><strong>Notes:</strong></p>\n<ul>\n<li>powers can be entered using ^</li>\n<li>to divide, use /</li>\n</ul>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-8..8 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "n": {"definition": "random(0..4#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "b": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "{a}*2^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}*2)-1)-{a}*1/({n}+1)^2*(({n}+1)*ln({b})-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"pickingStrategy": "all-ordered", "name": "Partial fractions", "pickQuestions": 1, "questions": [{"name": "Integrate with partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "xmc", "xmd"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>Factorise the denominator: $\\simplify[all]{x^2+{c+d}*x+{c*d}}=$ [[0]]</p>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>You should factorise and will not need to use powers.</p>", "showStrings": false, "strings": ["^"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "(x+{c})*(x+{d})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"maxAnswers": 0, "displayColumns": 0, "prompt": "<p>Select the correct form of the partial fraction:</p>", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "minAnswers": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "maxMarks": 0, "choices": ["<p>\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d}}}\\]</p>", "<p>\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d+1}}}\\]</p>", "<p>\\[\\frac{A}{\\simplify{x+{c-1}}}+\\frac{B}{\\simplify{x+{d}}}\\]</p>", "<p>\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x-{d}}}\\]</p>"], "type": "1_n_2", "distractors": ["", "", "", ""], "minMarks": 0}, {"prompt": "<p>Cross-multiply: (Note:You will need to use * for multiplication.)</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; border-bottom: solid;\">$A$</td>\n<td>$+$</td>\n<td style=\"text-align: center; border-bottom: solid;\">$B$</td>\n<td>$=$</td>\n<td style=\"text-align: center; border-bottom: solid;\">[[0]]</td>\n</tr>\n<tr>\n<td>$\\simplify{x+{c}}$</td>\n<td></td>\n<td>$\\simplify{x+{d}}$</td>\n<td></td>\n<td>$\\simplify{(x+{c})*(x+{d})}$</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "A*(x+{d})+B*(x+{c})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Equate the numerators of the original fraction $\\frac{\\simplify[all]{{a+b}*x+{a*d+b*c}}}{\\simplify[all]{x^2+{c+d}*x+{c*d}}}$ and your fraction from step c):</p>\n<p style=\"text-align: center;\">[[0]] $=$ [[1]]</p>\n<p></p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a+b}*x+{a*d+b*c}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "A*(x+{d})+B*(x+{c})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Substitute $x=\\var{-c}$ into the numerators: [[0]] = [[1]]</p>\n<p>Therefore $A=$ [[2]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{xmc}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({d}-{c})*A", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Substitute $x=\\var{-d}$ into the numerators: [[0]] = [[1]]</p>\n<p>Therefore $B=$ [[2]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{xmd}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({c}-{d})*B", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Therefore&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; border-bottom: solid;\">$\\simplify[all]{{a+b}*x+{a*d+b*c}}$</td>\n<td>$=$</td>\n<td style=\"text-align: center; border-bottom: solid;\">[[0]]</td>\n<td>$+$</td>\n<td style=\"text-align: center; border-bottom: solid;\">[[1]]</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\simplify[all]{x^2+{c+d}*x+{c*d}}$</td>\n<td></td>\n<td style=\"text-align: center;\">$\\simplify{x+{c}}$</td>\n<td></td>\n<td style=\"text-align: center;\">$\\simplify{x+{d}}$</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "<p>Hence calculate the following indefinite integral:</p>\n<p>\\[\\int\\frac{\\simplify[all]{{a+b}*x+{a*d+b*c}}}{\\simplify[all]{x^2+{c+d}*x+{c*d}}}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*ln(x+{c})+{b}*ln(x+{d})+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral can be integrated using&nbsp;<span style=\"text-decoration: underline;\">partial fractions</span>:</p>\n<p>\\[\\int\\frac{\\simplify[all]{{a+b}*x+{a*d+b*c}}}{\\simplify[all]{x^2+{c+d}*x+{c*d}}}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-4..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-4..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-4..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-4..4 except c)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "xmd": {"definition": "-{b}*{d}+{b}*{c}", "templateType": "anything", "group": "Ungrouped variables", "name": "xmd", "description": ""}, "xmc": {"definition": "-{a}*c+{a}*{d}", "templateType": "anything", "group": "Ungrouped variables", "name": "xmc", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integrate with partial fractions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "ans", "n1", "n2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"maxAnswers": 0, "displayColumns": 0, "prompt": "<p>Select the correct form of the partial fraction:</p>", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "minAnswers": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "maxMarks": 0, "choices": ["<p>\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d}}}\\]</p>", "<p>\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d+1}}}\\]</p>", "<p>\\[\\frac{A}{\\simplify{x+{c-1}}}+\\frac{B}{\\simplify{x+{d}}}\\]</p>", "<p>\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x-{d}}}\\]</p>"], "type": "1_n_2", "distractors": ["", "", "", ""], "minMarks": 0}, {"prompt": "<p>Therefore, by cross-multiplication:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; border-bottom: solid;\"><span style=\"text-align: start;\">$\\simplify[all]{{a}*x+{b}}$</span></td>\n<td>$=$</td>\n<td style=\"text-align: center; border-bottom: solid;\">$A$</td>\n<td>$+$</td>\n<td style=\"text-align: center; border-bottom: solid;\">$B$</td>\n<td>$=$</td>\n<td style=\"text-align: center; border-bottom: solid;\">[[0]]</td>\n</tr>\n<tr>\n<td>$\\simplify{(x+{c})*(x+{d})}$</td>\n<td></td>\n<td>$\\simplify{x+{c}}$</td>\n<td></td>\n<td>$\\simplify{x+{d}}$</td>\n<td></td>\n<td style=\"text-align: center;\">$\\simplify{(x+{c})*(x+{d})}$</td>\n</tr>\n</tbody>\n</table>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "A*(x+{d})+B*(x+{c})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Substitute $x=\\var{-c}$ into the numerators above: [[0]] = [[1]]</p>\n<p>Therefore $A=$ [[2]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{b-a*c}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({d}-{c})*A", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{n1}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Substitute $x=\\var{-d}$ into the numerators: [[0]] = [[1]]</p>\n<p>Therefore $B=$ [[2]]</p>", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{b-a*d}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({c}-{d})*B", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{n2}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "<p>Hence calculate the following indefinite integral:</p>\n<p>\\[\\int\\frac{\\simplify[all]{{a}*x+{b}}}{\\simplify{(x+{c})*(x+{d})}}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{n1}*ln(x+{c})+{n2}*ln(x+{d})+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 2 significant figures:</p>\n<p>\\[\\int_1^{10}\\frac{\\simplify[all]{{a}*x+{b}}}{\\simplify{(x+{c})*(x+{d})}}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "2", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "sigfig", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral can be integrated using&nbsp;<span style=\"text-decoration: underline;\">partial fractions</span>:</p>\n<p>\\[\\int\\frac{\\simplify[all]{{a}*x+{b}}}{\\simplify{(x+{c})*(x+{d})}}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": "15", "condition": ""}, "variables": {"a": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(0..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-5..5 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "c+random(3,4,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "ans": {"definition": "{n1}*(ln(10+{c})-ln(1+{c}))+{n2}*(ln(10+{d})-ln(1+{d}))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "n1": {"definition": "(b-a*c)/(d-c)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "(b-a*d)/(c-d)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"pickingStrategy": "all-ordered", "name": "Trig product", "pickQuestions": 1, "questions": [{"name": "Integrate trig product", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>Rewrite $\\simplify{{a}*sin({b}x)*cos({c}x)}$ using one of the trigonometric product identities:</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/2*(sin(({b}+{c})*x)+sin(({b}-{c})*x))", "steps": [{"showCorrectAnswer": true, "prompt": "<p>The 4 trigonometric product identities are:</p>\n<p>\\begin{eqnarray}\\sin(A)\\cos(B)&amp;&amp;=\\frac{1}{2}[\\sin(A+B)+\\sin(A-B)]\\\\<br/>\\cos(A)\\sin(B)&amp;&amp;=\\frac{1}{2}[\\sin(A+B)-\\sin(A-B)]\\\\<br/>\\cos(A)\\cos(B)&amp;&amp;=\\frac{1}{2}[\\cos(A+B)+\\cos(A-B)]\\\\<br/>\\sin(A)\\sin(B)&amp;&amp;=-\\frac{1}{2}[\\cos(A+B)-\\cos(A-B)]\\end{eqnarray}</p>", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}*sin({b}x)*cos({c}x)}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/(2({b}+{c}))*cos(({b}+{c})*x)-{a}/(2({b}-{c}))*cos(({b}-{c})*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 1 decimal place:</p>\n<p>\\[\\int^{\\pi/2}_0\\simplify{{a}*sin({b}x)*cos({c}x)}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "1", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral involves a&nbsp;<span style=\"text-decoration: underline;\">trigonometric product</span>:</p>\n<p>\\[\\int\\simplify{{a}*sin({b}x)*cos({c}x)}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "2*random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(c+1..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "-{a}/(2({b}+{c}))*cos(({b}+{c})*pi/2)-{a}/(2({b}-{c}))*cos(({b}-{c})*pi/2)+{a}/(2({b}+{c}))+{a}/(2({b}-{c}))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integrate trig product 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>Rewrite $\\simplify{{a}*sin({b}x)*sin({c}x)}$ using one of the trigonometric product identities:</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/2*(cos(({b}+{c})*x)-cos(({b}-{c})*x))", "steps": [{"showCorrectAnswer": true, "prompt": "<p>The 4 trigonometric product identities are:</p>\n<p>\\begin{eqnarray}\\sin(A)\\cos(B)&amp;&amp;=\\frac{1}{2}[\\sin(A+B)+\\sin(A-B)]\\\\<br/>\\cos(A)\\sin(B)&amp;&amp;=\\frac{1}{2}[\\sin(A+B)-\\sin(A-B)]\\\\<br/>\\cos(A)\\cos(B)&amp;&amp;=\\frac{1}{2}[\\cos(A+B)+\\cos(A-B)]\\\\<br/>\\sin(A)\\sin(B)&amp;&amp;=-\\frac{1}{2}[\\cos(A+B)-\\cos(A-B)]\\end{eqnarray}</p>", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Hence write down the indefinite integral:</p>\n<p>\\[\\int\\simplify{{a}*sin({b}x)*sin({c}x)}\\,dx\\]</p>\n<p>(Don't forget the constant of integration as this is an indefinite integral.)</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/(2({b}+{c}))*sin(({b}+{c})*x)+{a}/(2({b}-{c}))*sin(({b}-{c})*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Evaluate the definite integral, correct to 2 significant figures:</p>\n<p>\\[\\int_{-\\pi/3}^{\\pi/3}\\simplify{{a}*sin({b}x)*sin({c}x)}\\,dx\\]</p>", "expectedvariablenames": [], "checkingaccuracy": "2", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "sigfig", "checkvariablenames": false, "type": "jme"}], "statement": "<p>The following integral involves a&nbsp;<span style=\"text-decoration: underline;\">trigonometric product</span>:</p>\n<p>\\[\\int\\simplify{{a}*sin({b}x)*sin({c}x)}\\,dx\\]</p>", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(3..15)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(c+1..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "-{a}/({b}+{c})*sin(({b}+{c})*pi/3)+{a}/({b}-{c})*sin(({b}-{c})*pi/3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "showQuestionGroupNames": false, "metadata": {"description": "<p>Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "exam", "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Rob Beckett", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2450/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2642/"}, {"name": "Hans Fetter", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4478/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], 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