What is the integral of (x^5)/(x^3+8) ?

The integral of (x^5)/(x^3+8) is (x^3)/3-8/3 ln|x^3+8|+C

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integral of (x^5)/(x^3+8)

\int \frac{x ^{5}}{x ^{3} + 8} d x

\frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8 + C

Solution steps

\int \frac{x ^{5}}{x ^{3} + 8} d x

Long division

\frac{x ^{5}}{x ^{3} + 8} : x^{2} - \frac{8 x ^{2}}{x ^{3} + 8}

= \int x^{2} - \frac{8 x ^{2}}{x ^{3} + 8} 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^{2} d x - \int \frac{8 x ^{2}}{x ^{3} + 8} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{8 x ^{2}}{x ^{3} + 8} d x = \frac{8}{3} ln x^{3} + 8

= \frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8 + C

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What is the integral of (x^5)/(x^3+8) ?
The integral of (x^5)/(x^3+8) is (x^3)/3-8/3 ln|x^3+8|+C