What is the integral of (\sqrt[5]{x}+3x^{-2})^2 ?

The integral of (\sqrt[5]{x}+3x^{-2})^2 is -3/(x^3)-(15)/(2x^{4/5)}+5/7 x^{7/5}+C

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integral of (\sqrt[5]{x}+3x^{-2})^2

- \frac{3}{x ^{3}} - \frac{15}{2 x ^{\frac{4}{5}}} + \frac{5}{7} x^{\frac{7}{5}} + C

Solution steps

Expand

= \int \frac{9}{x ^{4}} + \frac{6}{x ^{\frac{9}{5}}} + x^{\frac{2}{5}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{9}{x ^{4}} d x + \int \frac{6}{x ^{\frac{9}{5}}} d x + \int x^{\frac{2}{5}} d x

\int \frac{9}{x ^{4}} d x = - \frac{3}{x ^{3}}

\int \frac{6}{x ^{\frac{9}{5}}} d x = - \frac{15}{2 x ^{\frac{4}{5}}}

\int x^{\frac{2}{5}} d x = \frac{5}{7} x^{\frac{7}{5}}

= - \frac{3}{x ^{3}} - \frac{15}{2 x ^{\frac{4}{5}}} + \frac{5}{7} x^{\frac{7}{5}}

Add a constant to the solution

= - \frac{3}{x ^{3}} - \frac{15}{2 x ^{\frac{4}{5}}} + \frac{5}{7} x^{\frac{7}{5}} + C

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What is the integral of (\sqrt[5]{x}+3x^{-2})^2 ?
The integral of (\sqrt[5]{x}+3x^{-2})^2 is -3/(x^3)-(15)/(2x^{4/5)}+5/7 x^{7/5}+C