What is the integral of (x-4)(x^2-8x)^3 ?

The integral of (x-4)(x^2-8x)^3 is (x^8)/8-4x^7+48x^6-256x^5+512x^4+C

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integral of (x-4)(x^2-8x)^3

\int (x - 4) (x^{2} - 8 x)^{3} d x

\frac{x ^{8}}{8} - 4 x^{7} + 48 x^{6} - 256 x^{5} + 512 x^{4} + C

Solution steps

\int (x - 4) (x^{2} - 8 x)^{3} d x

Expand

(x - 4) (x^{2} - 8 x)^{3} : x^{7} - 28 x^{6} + 288 x^{5} - 1280 x^{4} + 2048 x^{3}

= \int x^{7} - 28 x^{6} + 288 x^{5} - 1280 x^{4} + 2048 x^{3} 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^{7} d x - \int 28 x^{6} d x + \int 288 x^{5} d x - \int 1280 x^{4} d x + \int 2048 x^{3} d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 28 x^{6} d x = 4 x^{7}

\int 288 x^{5} d x = 48 x^{6}

\int 1280 x^{4} d x = 256 x^{5}

\int 2048 x^{3} d x = 512 x^{4}

= \frac{x ^{8}}{8} - 4 x^{7} + 48 x^{6} - 256 x^{5} + 512 x^{4}

Add a constant to the solution

= \frac{x ^{8}}{8} - 4 x^{7} + 48 x^{6} - 256 x^{5} + 512 x^{4} + C

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What is the integral of (x-4)(x^2-8x)^3 ?
The integral of (x-4)(x^2-8x)^3 is (x^8)/8-4x^7+48x^6-256x^5+512x^4+C