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 $\\displaystyle \\simplify[std]{((x +{a})*(x+{b}))}$ we obtain:
$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}*x+{d}} \\Rightarrow \\simplify[std]{(A+B)*x+{b}*A+{a}*B={c}*x+{d}}$
\nOne way to find A and B is to compare coefficients:
\nIdentifying coefficients:
\nConstant term: $\\simplify[std]{{b}*A+{a}*B = {d}}$
\nCoefficient of $x$: $ \\simplify[std]{A+B={c}}$ which gives $A =\\var{c} -B$
\nOn solving these simultaneous 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*}\\]
", "name": "Simon's copy of Simon's copy of Simon's copy of Integration by partial fractions. (Video)", "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "d", "d1", "s2", "s3", "b1", "s1", "b", "c"], "variable_groups": [], "variables": {"s2": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2", "templateType": "anything"}, "s1": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1", "templateType": "anything"}, "b1": {"group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "b1", "templateType": "anything"}, "d1": {"group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "d1", "templateType": "anything"}, "c": {"group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "c", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "definition": "if(d1=a*c,if(d1+1=b*c,d1+2,d1+1),if(d1=b*c,if(d1+1=a*c,d1+2,d1+1),d1))", "description": "", "name": "d", "templateType": "anything"}, "s3": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "definition": "if(b1=a,b1+s3,b1)", "description": "", "name": "b", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a", "templateType": "anything"}}, "parts": [{"type": "gapfill", "sortAnswers": false, "steps": [{"type": "information", "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "First 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 $B$.
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"}, "unitTests": [], "customMarkingAlgorithm": "", "vsetRangePoints": 5, "checkVariableNames": false, "variableReplacements": [], "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "vsetRange": [11, 12], "failureRate": 1, "scripts": {}}], "stepsPenalty": 1, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "$I=\\;$[[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.
", "scripts": {}}], "extensions": [], "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 $C$.
\n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "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.
\nVideo in Show steps.
"}, "functions": {}, "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}