What is the integral of p(p+1)^5 ?

The integral of p(p+1)^5 is (p^7)/7+(5p^6)/6+2p^5+(5p^4)/2+(5p^3)/3+(p^2)/2+C

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integral of p(p+1)^5

\int p (p + 1)^{5} d p

\frac{p ^{7}}{7} + \frac{5 p ^{6}}{6} + 2 p^{5} + \frac{5 p ^{4}}{2} + \frac{5 p ^{3}}{3} + \frac{p ^{2}}{2} + C

Solution steps

\int p (p + 1)^{5} d p

Expand

p (p + 1)^{5} : p^{6} + 5 p^{5} + 10 p^{4} + 10 p^{3} + 5 p^{2} + p

= \int p^{6} + 5 p^{5} + 10 p^{4} + 10 p^{3} + 5 p^{2} + 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 5 p^{5} d p + \int 10 p^{4} d p + \int 10 p^{3} d p + \int 5 p^{2} d p + \int p d p

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

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

\int 10 p^{4} d p = 2 p^{5}

\int 10 p^{3} d p = \frac{5 p ^{4}}{2}

\int 5 p^{2} d p = \frac{5 p ^{3}}{3}

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

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{6} + 2 p^{5} + \frac{5 p ^{4}}{2} + \frac{5 p ^{3}}{3} + \frac{p ^{2}}{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{6} + 2 p^{5} + \frac{5 p ^{4}}{2} + \frac{5 p ^{3}}{3} + \frac{p ^{2}}{2} + C

\frac{\partial}{\partial y} (y - \frac{1}{x ^{2}})

\int tan (x) sec (x) d x

\frac{d y}{d t} = e^{- t}

\frac{\partial}{\partial y} (2 x z)

e^{- 4}

What is the integral of p(p+1)^5 ?
The integral of p(p+1)^5 is (p^7)/7+(5p^6)/6+2p^5+(5p^4)/2+(5p^3)/3+(p^2)/2+C