What is the integral of p(p+2)^7 ?

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

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integral of p(p+2)^7

\int p (p + 2)^{7} d p

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

Solution steps

\int p (p + 2)^{7} d p

Expand

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

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

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

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

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

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

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

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

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

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

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

Add a constant to the solution

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

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