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

The integral of p(p+6)^5 is (p^7)/7+5p^6+72p^5+540p^4+2160p^3+3888p^2+C

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integral of p(p+6)^5

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

\frac{p ^{7}}{7} + 5 p^{6} + 72 p^{5} + 540 p^{4} + 2160 p^{3} + 3888 p^{2} + C

Solution steps

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

Expand

p (p + 6)^{5} : p^{6} + 30 p^{5} + 360 p^{4} + 2160 p^{3} + 6480 p^{2} + 7776 p

= \int p^{6} + 30 p^{5} + 360 p^{4} + 2160 p^{3} + 6480 p^{2} + 7776 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 30 p^{5} d p + \int 360 p^{4} d p + \int 2160 p^{3} d p + \int 6480 p^{2} d p + \int 7776 p d p

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

\int 30 p^{5} d p = 5 p^{6}

\int 360 p^{4} d p = 72 p^{5}

\int 2160 p^{3} d p = 540 p^{4}

\int 6480 p^{2} d p = 2160 p^{3}

\int 7776 p d p = 3888 p^{2}

= \frac{p ^{7}}{7} + 5 p^{6} + 72 p^{5} + 540 p^{4} + 2160 p^{3} + 3888 p^{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + 5 p^{6} + 72 p^{5} + 540 p^{4} + 2160 p^{3} + 3888 p^{2} + C

\int \frac{2}{1 - 4 x ^{2}} d x

f (x) = 2 x - x^{2}, at x = 1

\frac{d}{d x} (\frac{x ^{3} - 1}{ln ( x )})

\int cos (x) sin (x) 1 + cos^{2} (x) d x

4 y^{^{''}} - 16 y = 0

What is the integral of p(p+6)^5 ?
The integral of p(p+6)^5 is (p^7)/7+5p^6+72p^5+540p^4+2160p^3+3888p^2+C