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

f (x) = - 4.9 x^{2} + 35 x + 4

\frac{x}{x - 2}

\frac{d}{d x} (x^{2} e^{x y})

\int \frac{e ^{3 x}}{e ^{3 x} + 6} d x

\frac{d}{d θ} (a sin (θ))

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