$I=\\;$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tClick on Show steps for help if you need it. You will lose 1 mark if you do so.
\n\t\t\t", "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": 0.0001, "vsetrangepoints": 5, "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 3, "answer": "({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C", "vsetrange": [11, 12]}], "stepsPenalty": 1, "marks": 0, "steps": [{"showCorrectAnswer": true, "prompt": "\n\t\t\t\t\tFirst of all factorise the denominator.
\n\t\t\t\t\tYou have to find $a$ and $b$ such that $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+a)*(x+b)}$.
\n\t\t\t\t\tThen use partial fractions to write:
\\[\\simplify[std]{{c}/((x +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]
for suitable integers or fractions $A$ and $B$.
\n\t\t\t\t\t", "marks": 0, "type": "information", "scripts": {}}]}], "advice": "\n\tFirst we factorise $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+{a})*(x+{b})}$.
\n\tYou can do this by spotting the factors or by completing the square.
\n\tNext we use partial fractions to find $A$ and $B$ such that:
\\[\\simplify[std]{{c}/((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:
$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}}\\Rightarrow \\simplify[std]{(A+B)x+{b}A+{a}B={c}}$
\n\tIdentifying coefficients:
\n\tConstant term: $\\simplify[std]{{b}*A+{a}*B = {c}}$
\n\tCoefficent $x$: $ \\simplify[std]{A + B = 0}$ which gives $A = -B$.
\n\tHence we obtain $\\displaystyle \\simplify[std]{A = {c}/{b-a}}$ and $\\displaystyle \\simplify[std]{B={-c}/{b-a}}$
\n\tWhich gives: \\[\\simplify[std]{{c}/((x +{a})*(x+{b})) = ({c}/{b-a})*(1/(x+{a}) -1/(x+{b}))}\\]
\n\tSo \\[\\begin{eqnarray*} I &=& \\simplify[std]{Int({c}/(x^2+{a+b}*x+{a*b}),x )}\\\\ &=& \\simplify[std]{Int({c}/((x +{a})*(x+{b})),x )}\\\\ &=& \\simplify[std]{({c}/{b-a})*(Int(1/(x+{a}),x) -Int(1/(x+{b}),x))}\\\\ &=& \\simplify[std]{({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C} \\end{eqnarray*}\\]
\n\t", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n\t\tFactorise $x^2+bx+c$ into 2 distinct linear factors and then find $\\displaystyle \\int \\frac{a}{x^2+bx+c }\\;dx$ using partial fractions or otherwise.
\n\t\t\n\t\t", "notes": "\n\t\t
5/08/2012:
\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", "factorizing a quadratic", "indefinite integration", "integration", "partial fractions", "two distinct linear factors"], "question_groups": [{"pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered", "name": ""}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n\tFind the following integral.
\n\t\\[I = \\simplify[std]{Int({c}/(x^2+{a+b}*x+{a*b}),x )}\\]
\n\tInput all numbers as fractions or integers and not decimals.
\n\tInput the constant of integration as $C$.
\n\t", "preamble": {"js": "", "css": ""}, "type": "question", "functions": {}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "b1"], "variables": {"c": {"description": "", "definition": "random(2..9)", "group": "Ungrouped variables", "templateType": "anything", "name": "c"}, "s2": {"description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "name": "s2"}, "b1": {"description": "", "definition": "s2*random(1..9)", "group": "Ungrouped variables", "templateType": "anything", "name": "b1"}, "s1": {"description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "name": "s1"}, "s3": {"description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "name": "s3"}, "a": {"description": "", "definition": "s1*random(1..9)", "group": "Ungrouped variables", "templateType": "anything", "name": "a"}, "b": {"description": "", "definition": "if(b1=a,b1+s3,b1)", "group": "Ungrouped variables", "templateType": "anything", "name": "b"}}, "variable_groups": [], "showQuestionGroupNames": false, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}