$I=\\;$[[0]]
\nInput all numbers as fractions or integers and not decimals.
\nInput the constant of integration as $C$.
\nClick on Show steps for help if you need it. You will lose 1 mark if you do so.
\n ", "type": "gapfill", "gaps": [{"showpreview": true, "notallowed": {"message": "Input all numbers as fractions or integers and not decimals.
", "partialCredit": 0, "strings": ["."], "showStrings": false}, "answersimplification": "std", "showCorrectAnswer": true, "checkvariablenames": false, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "vsetrangepoints": 5, "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 3, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "vsetrange": [11, 12]}], "stepsPenalty": 1, "marks": 0, "steps": [{"showCorrectAnswer": true, "prompt": "\nFirst of all factorise the denominator.
\nYou have to find $a$ and $b$ such that $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+a)*(x+b)}$
\nThen use partial fractions to write:
\\[\\simplify[std]{({c}*x+{d})/((x +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]
for suitable integers or fractions $A$ and $B$.
\n ", "marks": 0, "type": "information", "scripts": {}}]}], "advice": "First we factorise $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+{a})*(x+{b})}$. You can do this by spotting the factors or by completing the square.
Next we use partial fractions to find $A$ and $B$ such that:
\\[\\displaystyle \\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = A/(x+{a})+B/(x+{b})}\\]
Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$ we obtain:
$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}*x+{d}} \\Rightarrow \\simplify[std]{(A+B)*x+{b}*A+{a}*B={c}*x+{d}}$
\nIdentifying coefficients:
\nConstant term: $\\simplify[std]{{b}*A+{a}*B = {d}}$
\nCoefficent $x$: $ \\simplify[std]{A+B={c}}$ which gives $A =\\var{c} -B$
\nOn solving these equations we obtain $\\displaystyle \\simplify[std]{A = {d-a*c}/{b-a}}$ and $\\displaystyle \\simplify[std]{B={d-b*c}/{a-b}}$
\nWhich gives: \\[\\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = ({d-a*c}/{b-a})*(1/(x+{a}) )+({d-b*c}/{a-b})*(1/(x+{b}))}\\]
\nSo \\[\\begin{eqnarray*} I &=& \\simplify[std]{Int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )}\\\\ &=&\\simplify[std]{Int(({c}*x+{d})/((x +{a})*(x+{b})),x )}\\\\ &=& \\simplify[std]{({d-a*c}/{b-a})*(Int(1/(x+{a}),x)) +({d-b*c}/{a-b})Int(1/(x+{b}),x)}\\\\ &=& \\simplify[std]{({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C} \\end{eqnarray*}\\]
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\\displaystyle \\int \\frac{ax+b}{x^2+cx+d}\\;dx,\\;a \\neq 0$ using partial fractions or otherwise.
", "notes": "\n \t\t5/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tAdded decimal point as forbidden string.
\n \t\tNote the checking range is chosen so that the arguments of the log terms are always positive - could have used abs - might be better?
\n \t\tImproved display of Advice.
\n \t\tAdded information about Show steps, also introduced penalty of 1 mark.
\n \t\tAdded !noLeadingMinus to ruleset std for display purposes.
\n \t\t"}, "name": "Maria's copy of Integration by partial fractions", "tags": ["2 distinct linear factors", "Calculus", "MAS1601", "Steps", "checked2015", "completing the square", "constant of integration", "factorising a quadratic", "indefinite integration", "integration", "logarithms", "partial fractions", "two distinct linear factors"], "question_groups": [{"pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered", "name": ""}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\nFind the following integral.
\n\\[I = \\simplify[std]{Int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )}\\]
\nInput all numbers as fractions or integers and not decimals.
\nInput the constant of integration as $C$.
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