// Numbas version: exam_results_page_options {"name": "Luis's copy of Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["s3", "c", "b", "m"], "name": "Luis's copy of Integration by parts", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "parts": [{"prompt": "\n
$I=\\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx $
You are given that \\[I=\\simplify[std]{({b}x+{c})^{m+1}/{b^2*(m+1)*(m+2)}*g(x)+C}\\]
For a polynomial $g(x)$. You have to find $g(x)$.
$g(x)=\\;$[[0]]
\nYou can get help by clicking on Show steps. You will lose 1 mark if you do.
\n ", "gaps": [{"checkvariablenames": false, "showpreview": true, "vsetrangepoints": 5, "answersimplification": "std", "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 3, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "{b*(m+1)}*x-{c}", "showCorrectAnswer": true, "vsetrange": [0, 1]}], "steps": [{"type": "information", "prompt": "\n \n \nThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
Also you need to know that for $n \\neq -1$:
\\[ \\int (ax+b)^n dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]
Part a
\nThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We 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: $\\simplify[std]{g(x)={b*(m+1)}*x-{c}}$.
Find the following indefinite integral.
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n \t\t3/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tGot rid of redundant instructions about inputting constant of integration.
\n \t\tGot rid of instruction re not inputting decimals - no restriction needed, so no forbidden strings.
\n \t\tPenalised use of steps, 1 mark. Added message to that effect.
\n \t\tImproved Advice display.
\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 $g(x)$.
"}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}