What is the integral of p(p+2)^5 ?

The integral of p(p+2)^5 is (p^7)/7+(5p^6)/3+8p^5+20p^4+(80p^3)/3+16p^2+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of p(p+2)^5

\int p (p + 2)^{5} d p

\frac{p ^{7}}{7} + \frac{5 p ^{6}}{3} + 8 p^{5} + 20 p^{4} + \frac{80 p ^{3}}{3} + 16 p^{2} + C

Solution steps

\int p (p + 2)^{5} d p

Expand

p (p + 2)^{5} : p^{6} + 10 p^{5} + 40 p^{4} + 80 p^{3} + 80 p^{2} + 32 p

= \int p^{6} + 10 p^{5} + 40 p^{4} + 80 p^{3} + 80 p^{2} + 32 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 10 p^{5} d p + \int 40 p^{4} d p + \int 80 p^{3} d p + \int 80 p^{2} d p + \int 32 p d p

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

\int 10 p^{5} d p = \frac{5 p ^{6}}{3}

\int 40 p^{4} d p = 8 p^{5}

\int 80 p^{3} d p = 20 p^{4}

\int 80 p^{2} d p = \frac{80 p ^{3}}{3}

\int 32 p d p = 16 p^{2}

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{3} + 8 p^{5} + 20 p^{4} + \frac{80 p ^{3}}{3} + 16 p^{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{3} + 8 p^{5} + 20 p^{4} + \frac{80 p ^{3}}{3} + 16 p^{2} + C

(sin (18 x))^{3}

\frac{d}{d x} ((4 - 3 x)^{9})

\frac{\partial}{\partial x} (z^{2} e^{x} cos (y))

y = (x^{3} + 2)^{2} (x^{5} + 4)^{4}

\frac{s + 1}{s + 2}

What is the integral of p(p+2)^5 ?
The integral of p(p+2)^5 is (p^7)/7+(5p^6)/3+8p^5+20p^4+(80p^3)/3+16p^2+C