// Numbas version: exam_results_page_options
{"name": "Indefinite integral 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indefinite integral 2", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$</p>"}, "extensions": [], "preamble": {"js": "", "css": ""}, "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", "variable_groups": [], "functions": {}, "variables": {"b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(2..5)"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(2..9)"}, "n": {"name": "n", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(3..5)"}, "d": {"name": "d", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1..9)"}}, "tags": [], "parts": [{"variableReplacements": [], "gaps": [{"variableReplacements": [], "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input all numbers as integers or fractions and not decimals.</p>"}, "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "unitTests": [], "scripts": {}, "vsetRange": [0, 1], "marks": 3, "vsetRangePoints": 5, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.0001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "reldiff", "customMarkingAlgorithm": ""}], "type": "gapfill", "useCustomName": false, "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>", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "ungrouped_variables": ["b", "n", "a", "d"], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"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": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"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/"}]}