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

The integral of p(p+5)^5 is (p^7)/7+(25p^6)/6+50p^5+(625p^4)/2+(3125p^3)/3+(3125p^2)/2+C

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integral of p(p+5)^5

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

\frac{p ^{7}}{7} + \frac{25 p ^{6}}{6} + 50 p^{5} + \frac{625 p ^{4}}{2} + \frac{3125 p ^{3}}{3} + \frac{3125 p ^{2}}{2} + C

Solution steps

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

Expand

p (p + 5)^{5} : p^{6} + 25 p^{5} + 250 p^{4} + 1250 p^{3} + 3125 p^{2} + 3125 p

= \int p^{6} + 25 p^{5} + 250 p^{4} + 1250 p^{3} + 3125 p^{2} + 3125 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 25 p^{5} d p + \int 250 p^{4} d p + \int 1250 p^{3} d p + \int 3125 p^{2} d p + \int 3125 p d p

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

\int 25 p^{5} d p = \frac{25 p ^{6}}{6}

\int 250 p^{4} d p = 50 p^{5}

\int 1250 p^{3} d p = \frac{625 p ^{4}}{2}

\int 3125 p^{2} d p = \frac{3125 p ^{3}}{3}

\int 3125 p d p = \frac{3125 p ^{2}}{2}

= \frac{p ^{7}}{7} + \frac{25 p ^{6}}{6} + 50 p^{5} + \frac{625 p ^{4}}{2} + \frac{3125 p ^{3}}{3} + \frac{3125 p ^{2}}{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + \frac{25 p ^{6}}{6} + 50 p^{5} + \frac{625 p ^{4}}{2} + \frac{3125 p ^{3}}{3} + \frac{3125 p ^{2}}{2} + C

x \to \infty lim (\frac{1}{2} x)

\frac{1}{3} x^{3} - 3 x^{2} + 10 x + 6, at x = 2

\frac{x d y}{d x} = y

y = x sin (\frac{6}{x})

What is the integral of p(p+5)^5 ?
The integral of p(p+5)^5 is (p^7)/7+(25p^6)/6+50p^5+(625p^4)/2+(3125p^3)/3+(3125p^2)/2+C