// Numbas version: finer_feedback_settings
{"name": "Indefinite integral 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$</p>"}, "parts": [{"gaps": [{"answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "answerSimplification": "std", "checkingType": "reldiff", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Input all numbers as integers or fractions and not decimals.</p>"}, "showPreview": true, "marks": 3, "vsetRange": [0, 1], "scripts": {}, "unitTests": [], "vsetRangePoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "jme", "variableReplacements": [], "expectedVariableNames": [], "checkVariableNames": false, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "checkingAccuracy": 0.0001}], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "sortAnswers": false, "type": "gapfill", "prompt": "<p>$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[0]]</p>\n<p>Input all numbers as integers or fractions and not decimals. Remember to&nbsp;include the constant of integration $C$.</p>\n<p></p>", "extendBaseMarkingAlgorithm": true}], "advice": "<p>Let $u = \\simplify[std]{{a}*x+{d}}$. Then,<br/>\\[\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(u^{n})}.\\]</p>\n<p>Now,<br/>\\[\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(u^{n})} \\frac{dx}{du} du.\\]</p>\n<p>Rearrange $u = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(u-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{du} = \\frac{1}{\\var{a}}$.</p>\n<p>$\\displaystyle \\int \\frac{1}{u^n} du = -\\frac{1}{(n-1)u^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:</p>\n<p>\\[\\int \\simplify[std]{{b}/(u^{n})} \\frac{dx}{du} du = \\simplify[std]{{b}/({n-1}*u^{n-1})} \\cdot \\frac{1}{\\var{a}} + C = \\simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.\\]</p>", "tags": [], "extensions": [], "ungrouped_variables": ["b", "n", "a", "d"], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n\t  \n\t  \n\t  <p>Find the following indefinite integral.</p>\n\t  \n\t  \n\t  \n\t  <p>Input the constant of integration as $C$.</p>\n\t  \n\t  \n\t  \n\t", "preamble": {"css": "", "js": ""}, "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "functions": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}}, "name": "Indefinite integral 2", "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}