What is the integral of x(x-8)^3 ?

The integral of x(x-8)^3 is (x^5)/5-6x^4+64x^3-256x^2+C

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integral of x(x-8)^3

\int x (x - 8)^{3} d x

\frac{x ^{5}}{5} - 6 x^{4} + 64 x^{3} - 256 x^{2} + C

Solution steps

\int x (x - 8)^{3} d x

Expand

x (x - 8)^{3} : x^{4} - 24 x^{3} + 192 x^{2} - 512 x

= \int x^{4} - 24 x^{3} + 192 x^{2} - 512 x 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 24 x^{3} d x + \int 192 x^{2} d x - \int 512 x d x

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

\int 24 x^{3} d x = 6 x^{4}

\int 192 x^{2} d x = 64 x^{3}

\int 512 x d x = 256 x^{2}

= \frac{x ^{5}}{5} - 6 x^{4} + 64 x^{3} - 256 x^{2}

Add a constant to the solution

= \frac{x ^{5}}{5} - 6 x^{4} + 64 x^{3} - 256 x^{2} + C

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What is the integral of x(x-8)^3 ?
The integral of x(x-8)^3 is (x^5)/5-6x^4+64x^3-256x^2+C