// Numbas version: exam_results_page_options {"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration by substitution", "tags": [], "metadata": {"description": "
Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\nFind the following integral.
\nYou must input the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\nYou can click on Show steps to get a hint. You will lose 1 mark if you do so.
\nNote that $\\displaystyle \\int \\frac{1}{x}\\;dx=\\ln(|x|)+C$ and you must include the absolute value in the argument of $\\ln$. You input $|x|$ as abs(x).
\n ", "advice": "\nThis exercise is best solved by using substitution.
\nNote that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$
\nSo if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$
\nHence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$
\nHence the integral becomes:
\n\\[\\begin{eqnarray*} I&=&\\int\\;\\frac{du}{u}\\\\ &=&\\ln(|u|)+C\\\\ &=& \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]
A Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
\nThe using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]
\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]
\n$I=\\;$[[0]]
\nInput all numbers as integers or fractions.
\nDo not forget to include the constant of integration $C$.
\n ", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\nTry the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
\nNote that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$
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