What is the integral of x(x-2)^7 ?

The integral of x(x-2)^7 is (x^9)/9-(7x^8)/4+12x^7-(140x^6)/3+112x^5-168x^4+(448x^3)/3-64x^2+C

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integral of x(x-2)^7

\int x (x - 2)^{7} d x

\frac{x ^{9}}{9} - \frac{7 x ^{8}}{4} + 12 x^{7} - \frac{140 x ^{6}}{3} + 112 x^{5} - 168 x^{4} + \frac{448 x ^{3}}{3} - 64 x^{2} + C

Solution steps

\int x (x - 2)^{7} d x

Expand

x (x - 2)^{7} : x^{8} - 14 x^{7} + 84 x^{6} - 280 x^{5} + 560 x^{4} - 672 x^{3} + 448 x^{2} - 128 x

= \int x^{8} - 14 x^{7} + 84 x^{6} - 280 x^{5} + 560 x^{4} - 672 x^{3} + 448 x^{2} - 128 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^{8} d x - \int 14 x^{7} d x + \int 84 x^{6} d x - \int 280 x^{5} d x + \int 560 x^{4} d x - \int 672 x^{3} d x + \int 448 x^{2} d x - \int 128 x d x

\int x^{8} d x = \frac{x ^{9}}{9}

\int 14 x^{7} d x = \frac{7 x ^{8}}{4}

\int 84 x^{6} d x = 12 x^{7}

\int 280 x^{5} d x = \frac{140 x ^{6}}{3}

\int 560 x^{4} d x = 112 x^{5}

\int 672 x^{3} d x = 168 x^{4}

\int 448 x^{2} d x = \frac{448 x ^{3}}{3}

\int 128 x d x = 64 x^{2}

= \frac{x ^{9}}{9} - \frac{7 x ^{8}}{4} + 12 x^{7} - \frac{140 x ^{6}}{3} + 112 x^{5} - 168 x^{4} + \frac{448 x ^{3}}{3} - 64 x^{2}

Add a constant to the solution

= \frac{x ^{9}}{9} - \frac{7 x ^{8}}{4} + 12 x^{7} - \frac{140 x ^{6}}{3} + 112 x^{5} - 168 x^{4} + \frac{448 x ^{3}}{3} - 64 x^{2} + C

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What is the integral of x(x-2)^7 ?
The integral of x(x-2)^7 is (x^9)/9-(7x^8)/4+12x^7-(140x^6)/3+112x^5-168x^4+(448x^3)/3-64x^2+C