What is the integral of (x^5-8x)^2(5x^4-8) ?

The integral of (x^5-8x)^2(5x^4-8) is (x^{15})/3-8x^{11}+64x^7-(512x^3)/3+C

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integral of (x^5-8x)^2(5x^4-8)

\int (x^{5} - 8 x)^{2} (5 x^{4} - 8) d x

\frac{x ^{15}}{3} - 8 x^{11} + 64 x^{7} - \frac{512 x ^{3}}{3} + C

Solution steps

\int (x^{5} - 8 x)^{2} (5 x^{4} - 8) d x

Expand

(x^{5} - 8 x)^{2} (5 x^{4} - 8) : 5 x^{14} - 88 x^{10} + 448 x^{6} - 512 x^{2}

= \int 5 x^{14} - 88 x^{10} + 448 x^{6} - 512 x^{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 5 x^{14} d x - \int 88 x^{10} d x + \int 448 x^{6} d x - \int 512 x^{2} d x

\int 5 x^{14} d x = \frac{x ^{15}}{3}

\int 88 x^{10} d x = 8 x^{11}

\int 448 x^{6} d x = 64 x^{7}

\int 512 x^{2} d x = \frac{512 x ^{3}}{3}

= \frac{x ^{15}}{3} - 8 x^{11} + 64 x^{7} - \frac{512 x ^{3}}{3}

Add a constant to the solution

= \frac{x ^{15}}{3} - 8 x^{11} + 64 x^{7} - \frac{512 x ^{3}}{3} + C

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What is the integral of (x^5-8x)^2(5x^4-8) ?
The integral of (x^5-8x)^2(5x^4-8) is (x^{15})/3-8x^{11}+64x^7-(512x^3)/3+C