Find the following integral.
\nInput the constant of integration as $C$.
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\nChecked calculation. OK.
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\nAdded decimal point as forbidden string and included message in prompt about not entering decimals.
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Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
"}, "ungrouped_variables": ["a", "s1", "b", "m"], "parts": [{"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.
\nClick on Show steps to get further help. You will lose 1 mark if you do so.
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", "showStrings": false, "strings": ["."]}, "marks": 3, "checkingaccuracy": 0.001, "showpreview": true}], "stepsPenalty": 1, "scripts": {}, "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$
", "type": "information", "marks": 0, "scripts": {}, "showCorrectAnswer": true}]}], "variables": {"s1": {"definition": "random(1,-1)", "name": "s1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "m": {"definition": "random(4..9)", "name": "m", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"definition": "s1*random(1..9)", "name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(1..5)", "name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "type": "question", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "advice": "\n \n \nThis 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 \n \n \n\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n \n &=&\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n \n &=& \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n \n \\end{eqnarray*}\\]
\n \n \n \nA 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 $x$.