What is the integral of x(x+8)^6 ?

The integral of x(x+8)^6 is (x^8)/8+(48x^7)/7+160x^6+2048x^5+15360x^4+65536x^3+131072x^2+C

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integral of x(x+8)^6

\int x (x + 8)^{6} d x

\frac{x ^{8}}{8} + \frac{48 x ^{7}}{7} + 160 x^{6} + 2048 x^{5} + 15360 x^{4} + 65536 x^{3} + 131072 x^{2} + C

Solution steps

\int x (x + 8)^{6} d x

Expand

x (x + 8)^{6} : x^{7} + 48 x^{6} + 960 x^{5} + 10240 x^{4} + 61440 x^{3} + 196608 x^{2} + 262144 x

= \int x^{7} + 48 x^{6} + 960 x^{5} + 10240 x^{4} + 61440 x^{3} + 196608 x^{2} + 262144 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^{7} d x + \int 48 x^{6} d x + \int 960 x^{5} d x + \int 10240 x^{4} d x + \int 61440 x^{3} d x + \int 196608 x^{2} d x + \int 262144 x d x

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

\int 48 x^{6} d x = \frac{48 x ^{7}}{7}

\int 960 x^{5} d x = 160 x^{6}

\int 10240 x^{4} d x = 2048 x^{5}

\int 61440 x^{3} d x = 15360 x^{4}

\int 196608 x^{2} d x = 65536 x^{3}

\int 262144 x d x = 131072 x^{2}

= \frac{x ^{8}}{8} + \frac{48 x ^{7}}{7} + 160 x^{6} + 2048 x^{5} + 15360 x^{4} + 65536 x^{3} + 131072 x^{2}

Add a constant to the solution

= \frac{x ^{8}}{8} + \frac{48 x ^{7}}{7} + 160 x^{6} + 2048 x^{5} + 15360 x^{4} + 65536 x^{3} + 131072 x^{2} + C

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What is the integral of x(x+8)^6 ?
The integral of x(x+8)^6 is (x^8)/8+(48x^7)/7+160x^6+2048x^5+15360x^4+65536x^3+131072x^2+C