// Numbas version: exam_results_page_options {"name": "Integration by substitution Exam Q3 2018", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration by substitution Exam Q3 2018", "variable_groups": [], "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "jme", "marks": "2", "checkingaccuracy": 0.001, "variableReplacements": [], "vsetrangepoints": 5, "showpreview": true, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "
Input all numbers as integers or fractions and not as decimals.
"}, "checkvariablenames": false, "showFeedbackIcon": true, "vsetrange": [0, 1], "scripts": {}, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "showCorrectAnswer": true, "expectedvariablenames": [], "answersimplification": "std", "checkingtype": "absdiff"}], "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]
\n$I=\\;$[[0]]
\nInput numbers in your answer as integers or fractions and not as decimals.
\n"}], "ungrouped_variables": ["a", "s1", "b", "m"], "variables": {"a": {"definition": "random(1..5)", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b": {"definition": "s1*random(1..9)", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "m": {"definition": "random(4..9)", "name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s1": {"definition": "random(1,-1)", "name": "s1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
"}, "statement": "\n\tFind the following integral.
\n\tInput the constant of integration as $C$.
\n\t \n\t", "preamble": {"js": "", "css": ""}, "extensions": [], "tags": [], "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 $x$.