What is the integral of x(x+8)^4 ?

The integral of x(x+8)^4 is (x^6)/6+(32x^5)/5+96x^4+(2048x^3)/3+2048x^2+C

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integral of x(x+8)^4

\int x (x + 8)^{4} d x

\frac{x ^{6}}{6} + \frac{32 x ^{5}}{5} + 96 x^{4} + \frac{2048 x ^{3}}{3} + 2048 x^{2} + C

Solution steps

\int x (x + 8)^{4} d x

Expand

x (x + 8)^{4} : x^{5} + 32 x^{4} + 384 x^{3} + 2048 x^{2} + 4096 x

= \int x^{5} + 32 x^{4} + 384 x^{3} + 2048 x^{2} + 4096 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^{5} d x + \int 32 x^{4} d x + \int 384 x^{3} d x + \int 2048 x^{2} d x + \int 4096 x d x

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

\int 32 x^{4} d x = \frac{32 x ^{5}}{5}

\int 384 x^{3} d x = 96 x^{4}

\int 2048 x^{2} d x = \frac{2048 x ^{3}}{3}

\int 4096 x d x = 2048 x^{2}

= \frac{x ^{6}}{6} + \frac{32 x ^{5}}{5} + 96 x^{4} + \frac{2048 x ^{3}}{3} + 2048 x^{2}

Add a constant to the solution

= \frac{x ^{6}}{6} + \frac{32 x ^{5}}{5} + 96 x^{4} + \frac{2048 x ^{3}}{3} + 2048 x^{2} + C

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What is the integral of x(x+8)^4 ?
The integral of x(x+8)^4 is (x^6)/6+(32x^5)/5+96x^4+(2048x^3)/3+2048x^2+C