// Numbas version: exam_results_page_options {"name": "cormac's copy of Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "\n\t

Find the following integral.

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Input the constant of integration as $C$.

\n\t \n\t", "tags": ["Calculus", "Steps", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "substitution"], "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "name": "", "questions": []}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "progress": "ready", "name": "cormac's copy of Julie's copy of Indefinite integral by substitution", "variable_groups": [], "parts": [{"stepspenalty": 1.0, "marks": 0.0, "prompt": "\n\t\t\t

\$I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\$

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$I=\\;$[[0]]

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Input numbers in your answer as integers or fractions and not as decimals.

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Click on Show steps to get further help. You will lose 1 mark if you do so.

\n\t\t\t \n\t\t\t", "type": "gapfill", "steps": [{"type": "information", "marks": 0.0, "prompt": "

Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$

"}], "gaps": [{"answersimplification": "std", "checkingaccuracy": 0.001, "type": "jme", "checkingtype": "absdiff", "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "notallowed": {"showstrings": false, "partialcredit": 0.0, "message": "

Input all numbers as integers or fractions and not as decimals.

", "strings": ["."]}, "marks": 3.0}]}], "variables": {"a": {"name": "a", "definition": "random(1..5)"}, "b": {"name": "b", "definition": "s1*random(1..9)"}, "m": {"name": "m", "definition": "random(4..9)"}, "s1": {"name": "s1", "definition": "random(1,-1)"}}, "advice": "\n\t \n\t \n\t

This exercise is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$
Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.

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Hence the integral becomes:

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\$\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n\t \n\t &=&\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n\t \n\t &=& \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n\t \n\t \\end{eqnarray*}\$

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A Useful Result
This example can be generalised.
Suppose \$I = \\int\\; f'(x)g(f(x))\\;dx\$
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\$I = \\int g(u)\\;du \$
And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.

\n\t \n\t \n\t \n\t", "showQuestionGroupNames": false, "metadata": {"description": "

Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$

", "notes": "\n\t\t \t\t

2/08/2012:

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Checked calculation. OK.

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Added decimal point as forbidden string and included message in prompt about not entering decimals.

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Got rid of a redundant ruleset.

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\n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "functions": {}, "extensions": [], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}]}]}], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}]}