// 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, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "\n\t
Find the following integral.
\n\tInput 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)}\\]
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput numbers in your answer as integers or fractions and not as decimals.
\n\t\t\tClick 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\tThis 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 $\\frac{1}{\\var{2*a}}du$.
Hence the integral becomes:
\n\t \n\t \n\t \n\t\\[\\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*}\\]
\n\t \n\t \n\t \n\tA 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 $x$.
Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
", "notes": "\n\t\t \t\t2/08/2012:
\n\t\t \t\tAdded tags.
\n\t\t \t\tAdded description.
\n\t\t \t\tChecked calculation. OK.
\n\t\t \t\tAdded information about Show steps in prompt content area.
\n\t\t \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n\t\t \t\tGot rid of a redundant ruleset.
\n\t\t \t\t\n\t\t \t\t
\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/"}]}