// Numbas version: exam_results_page_options
{"name": "Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Indefinite integral", "tags": ["Calculus", "Steps", "calculus", "constant of integration", "indefinite integration", "integrals", "integration", "integration by substitution", "standard integrals", "steps", "substitution"], "advice": "\n  <p>Let $y = \\simplify[std]{{a}*x+{d}}$. Then,<br/>\\[\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(y^{n})}.\\]</p>\n    <p>Now,<br/>\\[\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy.\\]</p>\n    <p>Rearrange $y = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(y-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{dy} = \\frac{1}{\\var{a}}$.</p>\n    <p>$\\displaystyle \\int \\frac{1}{y^n} dx = -\\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:</p>\n    <p>\\[\\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy = \\simplify[std]{{b}/({n-1}*y^{n-1})} \\cdot \\frac{1}{\\var{a}} + C = \\simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <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>Click on Show steps to get help. You will lose 1 mark by doing so.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "reldiff", "answersimplification": "std", "marks": 3.0, "answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "type": "jme"}], "steps": [{"prompt": "<p>&nbsp;\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "\n  \n  \n  <p>Find the following indefinite integral.</p>\n  \n  \n  \n  <p>Input the constant of integration as $C$.</p>\n  \n  \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "b": {"definition": "random(2..5)", "name": "b"}, "d": {"definition": "random(1..9)", "name": "d"}, "n": {"definition": "random(3..5)", "name": "n"}}, "metadata": {"notes": "\n    \t\t<p><strong>2/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Added decimal point to forbidden strings along with message to user re input of numbers.</p>\n    \t\t<p>Added a Step and message about Show steps included - losing 1 mark if used as it gives the formula for finding the integral. Increased marks to 3 for the question, so that can cope with losing a mark for using Show steps.</p>\n    \t\t<p>Changed accuracy setting to relative difference of 0.00001 as we have negative powers.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Added message in prompt &nbsp;about including the constant of integration.</p>\n    \t\t<p>Noted issue with steps-answer order and the messages/marks generated.</p>\n    \t\t<p>Changed numerator to the range 2..5.</p>\n    \t\t<p>Improved display in Advice.</p>\n    \t\t<p>&nbsp;</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}