// Numbas version: exam_results_page_options {"name": "Luis's copy of Integration of fraction with power in denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "m", "n", "r"], "name": "Luis's copy of Integration of fraction with power in denominator", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "parts": [{"prompt": "\n
$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$
\nYou are given that \\[I=\\simplify[std]{g(x)*({a}x+{d})^{1-n}}+C\\] for a polynomial $g(x)$.
\nYou have to find $g(x)$.
\n$g(x)=\\;$[[0]]
\nRemember to input all numbers as integers or fractions.
\n ", "gaps": [{"checkvariablenames": false, "showpreview": true, "vsetrangepoints": 5, "answersimplification": "std", "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "({-m}/{n-2})*x-{m*d*(n-1)+r*(n-2)}/{(n-2)*(n-1)*a}", "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "strings": ["."], "showStrings": false, "partialCredit": 0}}], "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "marks": 0}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "type": "question", "functions": {}, "variables": {"c": {"name": "c", "templateType": "anything", "description": "", "definition": "m*d+r", "group": "Ungrouped variables"}, "d": {"name": "d", "templateType": "anything", "description": "", "definition": "random(1..5)", "group": "Ungrouped variables"}, "s1": {"name": "s1", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "b": {"name": "b", "templateType": "anything", "description": "", "definition": "m*a", "group": "Ungrouped variables"}, "s2": {"name": "s2", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "m": {"name": "m", "templateType": "anything", "description": "", "definition": "s1*random(1..4)", "group": "Ungrouped variables"}, "a": {"name": "a", "templateType": "anything", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}, "r": {"name": "r", "templateType": "anything", "description": "", "definition": "s2*random(1..5)", "group": "Ungrouped variables"}, "n": {"name": "n", "templateType": "anything", "description": "", "definition": "3", "group": "Ungrouped variables"}}, "variable_groups": [], "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "showQuestionGroupNames": false, "tags": ["Calculus", "calculus", "checked2015", "indefinite integral", "indefinite integration", "integration", "integration by substitution", "integration of a rational polynomial", "MAS1601", "substitution"], "advice": "\nLet $y = \\simplify[std]{{a}*x+{d}}$.
\nThen $x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ and so we have the numerator $\\simplify[std]{{b}*x+{c}}$ becomes in terms of $y$:
\n$\\simplify[std]{{b}*x+{c} = {b}*1/{a}*(y-{d})+{c}= {m}y+{r}}$ and so
\n\\[\\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} = \\simplify[std]{({m}*y+{r})/(y^{n})={m}/y^{n-1}+{r}/y^{n}}\\]
\nNow,
\\[\\int \\simplify[std]{({b}x+{c})/({a}*x+{d})^{n}} dx = \\int \\left(\\simplify[std]{{m}/y^{n-1}+{r}/y^{n}} \\right)\\frac{dx}{dy} dy \\]
Since $\\displaystyle x = \\simplify[std]{(y-{d})/{a}}$ then $\\displaystyle \\frac{dx}{dy} = \\frac{1}{\\var{a}}$.
\nWe can now calculate the desired integral:
\n\\[ \\begin{eqnarray*} \\int \\left(\\simplify[std]{{m}/y^{n-1}+{r}/y^{n}}\\right) \\frac{dx}{dy} dy &=&\\frac{1}{\\var{a}}\\left(\\int \\simplify[std]{{m}/y^{n-1}}\\;dy+\\int \\simplify[std]{{r}/y^{n}}\\;dy \\right)\\\\ &=&\\frac{1}{\\var{a}}\\left(\\simplify[std]{{-m}/({n-2}*y^{n-2})+ {-r}/({n-1}*y^{n-1})}\\right) + C \\\\ &=& \\simplify[std]{(-{m})/({a*(n-2)}*({a}*x+{d})^{n-2})+(-{r})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}\\\\ &=&\\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{a*(n-2)})*({a}x+{d})-{r}/({a*(n-1)}))}\\\\ &=&\\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a})} \\end{eqnarray*} \\]
Hence \\[g(x)=\\simplify[std]{({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a}}\\]
Find the following indefinite integral.
\nInput all numbers as integers or fractions, not as decimals.
\n\n ", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n \t\t
3/7/2012:
Added tags
\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tImproved spacing and display.
\n \t\tGot rid of instruction about including constant of integration as not needed.
\n \t\tChecked calculation.
\n \t\t", "description": "$\\displaystyle \\int \\frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$ for a 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/"}]}