What is the integral of (x^2+5/x)^2 ?

The integral of (x^2+5/x)^2 is (x^5)/5+5x^2-(25)/x+C

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integral of (x^2+5/x)^2

\int (x^{2} + \frac{5}{x})^{2} d x

\frac{x ^{5}}{5} + 5 x^{2} - \frac{25}{x} + C

Solution steps

\int (x^{2} + \frac{5}{x})^{2} d x

Expand

(x^{2} + \frac{5}{x})^{2} : x^{4} + 10 x + \frac{25}{x ^{2}}

= \int x^{4} + 10 x + \frac{25}{x ^{2}} 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 x^{4} d x + \int 10 x d x + \int \frac{25}{x ^{2}} d x

\int x^{4} d x = \frac{x ^{5}}{5}

\int 10 x d x = 5 x^{2}

\int \frac{25}{x ^{2}} d x = - \frac{25}{x}

= \frac{x ^{5}}{5} + 5 x^{2} - \frac{25}{x}

Add a constant to the solution

= \frac{x ^{5}}{5} + 5 x^{2} - \frac{25}{x} + C

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What is the integral of (x^2+5/x)^2 ?
The integral of (x^2+5/x)^2 is (x^5)/5+5x^2-(25)/x+C