What is the integral of p(p+7)^5 ?

The integral of p(p+7)^5 is (p^7)/7+(35p^6)/6+98p^5+(1715p^4)/2+(12005p^3)/3+(16807p^2)/2+C

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integral of p(p+7)^5

\int p (p + 7)^{5} d p

\frac{p ^{7}}{7} + \frac{35 p ^{6}}{6} + 98 p^{5} + \frac{1715 p ^{4}}{2} + \frac{12005 p ^{3}}{3} + \frac{16807 p ^{2}}{2} + C

Solution steps

\int p (p + 7)^{5} d p

Expand

p (p + 7)^{5} : p^{6} + 35 p^{5} + 490 p^{4} + 3430 p^{3} + 12005 p^{2} + 16807 p

= \int p^{6} + 35 p^{5} + 490 p^{4} + 3430 p^{3} + 12005 p^{2} + 16807 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 35 p^{5} d p + \int 490 p^{4} d p + \int 3430 p^{3} d p + \int 12005 p^{2} d p + \int 16807 p d p

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

\int 35 p^{5} d p = \frac{35 p ^{6}}{6}

\int 490 p^{4} d p = 98 p^{5}

\int 3430 p^{3} d p = \frac{1715 p ^{4}}{2}

\int 12005 p^{2} d p = \frac{12005 p ^{3}}{3}

\int 16807 p d p = \frac{16807 p ^{2}}{2}

= \frac{p ^{7}}{7} + \frac{35 p ^{6}}{6} + 98 p^{5} + \frac{1715 p ^{4}}{2} + \frac{12005 p ^{3}}{3} + \frac{16807 p ^{2}}{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + \frac{35 p ^{6}}{6} + 98 p^{5} + \frac{1715 p ^{4}}{2} + \frac{12005 p ^{3}}{3} + \frac{16807 p ^{2}}{2} + C

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What is the integral of p(p+7)^5 ?
The integral of p(p+7)^5 is (p^7)/7+(35p^6)/6+98p^5+(1715p^4)/2+(12005p^3)/3+(16807p^2)/2+C