What is the integral of p(p+5)^6 ?

The integral of p(p+5)^6 is (p^8)/8+(30p^7)/7+(125p^6)/2+500p^5+(9375p^4)/4+6250p^3+(15625p^2)/2+C

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integral of p(p+5)^6

\int p (p + 5)^{6} d p

\frac{p ^{8}}{8} + \frac{30 p ^{7}}{7} + \frac{125 p ^{6}}{2} + 500 p^{5} + \frac{9375 p ^{4}}{4} + 6250 p^{3} + \frac{15625 p ^{2}}{2} + C

Solution steps

\int p (p + 5)^{6} d p

Expand

p (p + 5)^{6} : p^{7} + 30 p^{6} + 375 p^{5} + 2500 p^{4} + 9375 p^{3} + 18750 p^{2} + 15625 p

= \int p^{7} + 30 p^{6} + 375 p^{5} + 2500 p^{4} + 9375 p^{3} + 18750 p^{2} + 15625 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 30 p^{6} d p + \int 375 p^{5} d p + \int 2500 p^{4} d p + \int 9375 p^{3} d p + \int 18750 p^{2} d p + \int 15625 p d p

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

\int 30 p^{6} d p = \frac{30 p ^{7}}{7}

\int 375 p^{5} d p = \frac{125 p ^{6}}{2}

\int 2500 p^{4} d p = 500 p^{5}

\int 9375 p^{3} d p = \frac{9375 p ^{4}}{4}

\int 18750 p^{2} d p = 6250 p^{3}

\int 15625 p d p = \frac{15625 p ^{2}}{2}

= \frac{p ^{8}}{8} + \frac{30 p ^{7}}{7} + \frac{125 p ^{6}}{2} + 500 p^{5} + \frac{9375 p ^{4}}{4} + 6250 p^{3} + \frac{15625 p ^{2}}{2}

Add a constant to the solution

= \frac{p ^{8}}{8} + \frac{30 p ^{7}}{7} + \frac{125 p ^{6}}{2} + 500 p^{5} + \frac{9375 p ^{4}}{4} + 6250 p^{3} + \frac{15625 p ^{2}}{2} + C

What is the integral of p(p+5)^6 ?
The integral of p(p+5)^6 is (p^8)/8+(30p^7)/7+(125p^6)/2+500p^5+(9375p^4)/4+6250p^3+(15625p^2)/2+C