What is the integral of (x^8)/(x^3+2) ?

The integral of (x^8)/(x^3+2) is (x^6)/6-(2x^3)/3+4/3 ln|x^3+2|+C

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integral of (x^8)/(x^3+2)

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

\frac{x ^{6}}{6} - \frac{2 x ^{3}}{3} + \frac{4}{3} ln x^{3} + 2 + C

Solution steps

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

Long division

\frac{x ^{8}}{x ^{3} + 2} : x^{5} - 2 x^{2} + \frac{4 x ^{2}}{x ^{3} + 2}

= \int x^{5} - 2 x^{2} + \frac{4 x ^{2}}{x ^{3} + 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^{5} d x - \int 2 x^{2} d x + \int \frac{4 x ^{2}}{x ^{3} + 2} d x

\int x^{5} d x = \frac{x ^{6}}{6}

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

\int \frac{4 x ^{2}}{x ^{3} + 2} d x = \frac{4}{3} ln x^{3} + 2

= \frac{x ^{6}}{6} - \frac{2 x ^{3}}{3} + \frac{4}{3} ln x^{3} + 2

Add a constant to the solution

= \frac{x ^{6}}{6} - \frac{2 x ^{3}}{3} + \frac{4}{3} ln x^{3} + 2 + C

4 x^{4} + 3 x^{3} + 2 x^{2} + x + 1, 1

\frac{\partial}{\partial x} (- 5 y e^{x})

\frac{( s + 2 )}{s ( s ^{2} + 9 )}

\frac{d}{d x} (1 2^{x})

\frac{d}{d x} (x^{2} - 4 x + 5)

What is the integral of (x^8)/(x^3+2) ?
The integral of (x^8)/(x^3+2) is (x^6)/6-(2x^3)/3+4/3 ln|x^3+2|+C