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Section 5.2 : Computing Indefinite Integrals
6. Evaluate \( \displaystyle \int{{\sqrt[3]{w} + 10\,\,\sqrt[5]{{{w^3}}}\,dw}}\).
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Hint : Don’t forget to convert the roots to fractional exponents.
We first need to convert the roots to fractional exponents.
\[\int{{\sqrt[3]{w} + 10\,\,\sqrt[5]{{{w^3}}}\,dw}} = \int{{{w^{\,\frac{1}{3}}} + 10\,{{\left( {{w^3}} \right)}^{\frac{1}{5}}}\,dw}} = \int{{{w^{\,\frac{1}{3}}} + 10{w^{\,\frac{3}{5}}}\,dw}}\] Show Step 2Once we’ve gotten the roots converted to fractional exponents there really isn’t too much to do other than to evaluate the integral.
\[\int{{\sqrt[3]{w} + 10\,\,\sqrt[5]{{{w^3}}}\,dw}} = \int{{{w^{\,\frac{1}{3}}} + 10{w^{\,\frac{3}{5}}}\,dw}} = \frac{3}{4}{w^{\,\frac{4}{3}}} + 10\left( {\frac{5}{8}} \right){w^{\,\frac{8}{5}}} + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{3}{4}{w^{\,\frac{4}{3}}} + \frac{{25}}{4}{w^{\,\frac{8}{5}}} + c}}\]Don’t forget to add on the “+c” since we know that we are asking what function did we differentiate to get the integrand and the derivative of a constant is zero and so we do need to add that onto the answer.