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

The integral of p(p+6)^6 is (p^8)/8+(36p^7)/7+90p^6+864p^5+4860p^4+15552p^3+23328p^2+C

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integral of p(p+6)^6

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

\frac{p ^{8}}{8} + \frac{36 p ^{7}}{7} + 90 p^{6} + 864 p^{5} + 4860 p^{4} + 15552 p^{3} + 23328 p^{2} + C

Solution steps

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

Expand

p (p + 6)^{6} : p^{7} + 36 p^{6} + 540 p^{5} + 4320 p^{4} + 19440 p^{3} + 46656 p^{2} + 46656 p

= \int p^{7} + 36 p^{6} + 540 p^{5} + 4320 p^{4} + 19440 p^{3} + 46656 p^{2} + 46656 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 36 p^{6} d p + \int 540 p^{5} d p + \int 4320 p^{4} d p + \int 19440 p^{3} d p + \int 46656 p^{2} d p + \int 46656 p d p

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

\int 36 p^{6} d p = \frac{36 p ^{7}}{7}

\int 540 p^{5} d p = 90 p^{6}

\int 4320 p^{4} d p = 864 p^{5}

\int 19440 p^{3} d p = 4860 p^{4}

\int 46656 p^{2} d p = 15552 p^{3}

\int 46656 p d p = 23328 p^{2}

= \frac{p ^{8}}{8} + \frac{36 p ^{7}}{7} + 90 p^{6} + 864 p^{5} + 4860 p^{4} + 15552 p^{3} + 23328 p^{2}

Add a constant to the solution

= \frac{p ^{8}}{8} + \frac{36 p ^{7}}{7} + 90 p^{6} + 864 p^{5} + 4860 p^{4} + 15552 p^{3} + 23328 p^{2} + C

What is the integral of p(p+6)^6 ?
The integral of p(p+6)^6 is (p^8)/8+(36p^7)/7+90p^6+864p^5+4860p^4+15552p^3+23328p^2+C