// Numbas version: exam_results_page_options {"name": "Integration by parts - Ch 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["s3", "c", "b", "m"], "name": "Integration by parts - Ch 1", "tags": ["algebraic manipulation", "Calculus", "calculus", "indefinite integration", "integrals", "integration", "integration by parts", "steps", "Steps"], "advice": "
The formula for integrating by parts is
\n\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
\nWe choose $u = x$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{({b}*x+{c})^{m}}$.
\nSo $\\displaystyle \\frac{du}{dx}$ = $1$ and $\\displaystyle v = \\simplify[std]{(1/{(m+1)*b})*({b}*x+{c})^{m+1}}$.
\nHence,
\\[ \\begin{eqnarray*} \\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*b})*Int (({b}*x+{c})^{m+1}, x)} \\\\ &=& \\simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*(m+2)*b^2})*({b}*x+{c})^{m+2}+C} \\\\ &=&\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+2)}x - ({b}x+{c}))+C}\\\\ &=&\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}x - {c})+C} \\end{eqnarray*}\\]
The solution is: $\\simplify[std]{g(x)={b*(m+1)}*x-{c}}$.
$I=\\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx $
Leave your answer in the form of $\\frac{(ax+b)^d}{e}*g(x) + C$ with your constant of integration as \"+ C\".
\n$I=\\;$[[0]]
\n\nYou can get help by clicking on show steps.
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\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
Find the following indefinite integral.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "c": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "m": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}}, "metadata": {"notes": "\n \t\t \t\t3/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tAdded description.
\n \t\t \t\tChecked calculation. OK.
\n \t\t \t\tGot rid of redundant instructions about inputting constant of integration.
\n \t\t \t\tGot rid of instruction re not inputting decimals - no restriction needed, so no forbidden strings.
\n \t\t \t\tPenalised use of steps, 1 mark. Added message to that effect.
\n \t\t \t\tImproved Advice display.
\n \t\t \n \t\t", "description": "Given that $\\displaystyle \\int x({ax+b)^{m}} dx=\\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.
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