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{"name": "Simon's copy of Indefinite Integrals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "metadata": {"description": "<p>Indefinite Integrals</p>\n<p>rebel</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"vsetRange": [0, 1], "vsetRangePoints": 5, "failureRate": 1, "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p>$\\int(x^4-\\var{a}x^3+\\var{b}x-\\var{c})\\mathrm{dx}$</p>", "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "x^5/5-{a}x^4/4+{b}x^2/2-{c}x+C", "showPreview": true, "expectedVariableNames": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "marks": 1, "type": "jme", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"vsetRange": [0, 1], "vsetRangePoints": 5, "failureRate": 1, "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p>$\\int(u+\\var{d})(2u+\\var{f})\\mathrm{du}$</p>", "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "2u^3/3+u^2({f}+2{d})/2+{d}{f}u+C", "showPreview": true, "expectedVariableNames": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "marks": 1, "type": "jme", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"vsetRange": [0, 1], "vsetRangePoints": 5, "failureRate": 1, "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p>$\\int\\frac{\\sin(2x)}{\\sin(x)}\\mathrm{dx}$.</p>\n<p>Take care to include the brackets in the trigonometric expressions, i.e. write sin(x) rather than sinx.</p>", "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "2sin(x) + C", "showPreview": true, "expectedVariableNames": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "marks": 1, "type": "jme", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "functions": {}, "statement": "<p>Solve the following indefinite integrals, using $C$ to represent an unknown constant.</p>", "preamble": {"css": "", "js": ""}, "name": "Simon's copy of Indefinite Integrals", "variables": {"f": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "definition": "random(1..8 except d)"}, "d": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "definition": "random(1..8)"}, "a": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "definition": "random(2..9)"}, "b": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "definition": "random(2..9 except a)"}, "c": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "definition": "random(1..9 except a except b)"}}, "rulesets": {}, "advice": "<p><strong>(a)</strong></p>\n<p>$\\int(x^4-\\var{a}x^3+\\var{b}x-\\var{c})\\mathrm{dx}$</p>\n<p>Using $\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C$ we obtain</p>\n<p>$\\int(x^4-\\var{a}x^3+\\var{b}x-\\var{c})\\mathrm{dx} =&nbsp;\\simplify{x^5/5-{a/4}x^4+{b/2}x^2-{c}x+C}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>To find $\\int(u+\\var{d})(2u+\\var{f})\\mathrm{du}$ we can first multiply out the brackets to obtain:</p>\n<p>$\\int \\simplify{2u^2+{2d+f}u+{d*f}}\\;{du} = \\simplify{2/3u^3+{(2d+f)/2}u^2+{d*f}u+C} $</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>To find $\\int\\frac{\\sin(2x)}{\\sin(x)}\\mathrm{dx}$ we use the identity $\\sin(2x)=2\\sin(x)\\cos(x)$</p>\n<p>Hence our integral becomes&nbsp;$\\int\\frac{2\\sin(x)\\cos(x)}{\\sin(x)}\\mathrm{dx}=\\int2\\cos(x)\\mathrm{dx}=2\\sin(x)+C$</p>", "ungrouped_variables": ["a", "c", "b", "d", "f"], "extensions": [], "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}