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

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

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integral of p(p+8)^6

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

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

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

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

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

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

Add a constant to the solution

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

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