What is the integral of p(p+8)^5 ?

The integral of p(p+8)^5 is (p^7)/7+(20p^6)/3+128p^5+1280p^4+(20480p^3)/3+16384p^2+C

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integral of p(p+8)^5

\int p (p + 8)^{5} d p

\frac{p ^{7}}{7} + \frac{20 p ^{6}}{3} + 128 p^{5} + 1280 p^{4} + \frac{20480 p ^{3}}{3} + 16384 p^{2} + C

Solution steps

\int p (p + 8)^{5} d p

Expand

p (p + 8)^{5} : p^{6} + 40 p^{5} + 640 p^{4} + 5120 p^{3} + 20480 p^{2} + 32768 p

= \int p^{6} + 40 p^{5} + 640 p^{4} + 5120 p^{3} + 20480 p^{2} + 32768 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^{6} d p + \int 40 p^{5} d p + \int 640 p^{4} d p + \int 5120 p^{3} d p + \int 20480 p^{2} d p + \int 32768 p d p

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

\int 40 p^{5} d p = \frac{20 p ^{6}}{3}

\int 640 p^{4} d p = 128 p^{5}

\int 5120 p^{3} d p = 1280 p^{4}

\int 20480 p^{2} d p = \frac{20480 p ^{3}}{3}

\int 32768 p d p = 16384 p^{2}

= \frac{p ^{7}}{7} + \frac{20 p ^{6}}{3} + 128 p^{5} + 1280 p^{4} + \frac{20480 p ^{3}}{3} + 16384 p^{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + \frac{20 p ^{6}}{3} + 128 p^{5} + 1280 p^{4} + \frac{20480 p ^{3}}{3} + 16384 p^{2} + C

What is the integral of p(p+8)^5 ?
The integral of p(p+8)^5 is (p^7)/7+(20p^6)/3+128p^5+1280p^4+(20480p^3)/3+16384p^2+C