// Numbas version: exam_results_page_options {"name": "Integration (Negative and Fractional Powers) 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration (Negative and Fractional Powers) 1", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "
Using the Table of Integrals/Antiderivatives, calculate the integral of $y=\\simplify[all,fractionNumbers]{{a}x^{b/c}}$.
\n", "advice": "From the Table of Integrals we see that a function of the form \\[ f(x)=x^n \\] has the integral \\[ \\int x^n dx = \\frac{x^{n+1}}{n+1}+ c,\\]
\nand \\[\\int kf(x) dx = k \\int f(x) dx.\\]
\nSo, for the function
\n\\[ y=\\simplify[fractionNumbers]{{a}x^{b/c}}, \\]
\nthe integral is
\n\\[ \\begin{split}\\simplify[fractionNumbers]{int({a}x^{b/c},x)} = \\simplify[fractionNumbers]{{a}int(x^{b/c},x)}&\\,= \\simplify[all,fractionNumbers,!collectNumbers,!simplifyFractions,!expandBrackets]{{a}({c/(b+c)}x^({b/c+1}))} + c,\\\\ \\\\&\\,= \\simplify[all,fractionNumbers]{{a/(b/c+1)}x^{b/c+1}}+c.\\end{split} \\]
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