// Numbas version: exam_results_page_options {"name": "Jo-Ann's Indefinite integrals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$

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Find the following indefinite integral.

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Input the constant of integration as $C$.

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Let $u = \\simplify[std]{{a}*x+{d}}$. Then,
\$\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(u^{n})}.\$

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Now,
\$\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(u^{n})} \\frac{dx}{du} du.\$

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Rearrange $u = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(u-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{du} = \\frac{1}{\\var{a}}$.

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$\\displaystyle \\int \\frac{1}{u^n} du = -\\frac{1}{(n-1)u^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:

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\$\\int \\simplify[std]{{b}/(u^{n})} \\frac{dx}{du} du = \\simplify[std]{{b}/({n-1}*u^{n-1})} \\cdot \\frac{1}{\\var{a}} + C = \\simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.\$

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Input all numbers as integers or fractions and not decimals.

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$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[0]]

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Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

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"}], "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}