What is the integral of x(2+sqrt(x))^2 ?

The integral of x(2+sqrt(x))^2 is 2x^2+8/5 x^{5/2}+(x^3)/3+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of x(2+sqrt(x))^2

\int x (2 + x)^{2} d x

2 x^{2} + \frac{8}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3} + C

Solution steps

\int x (2 + x)^{2} d x

Expand

x (2 + x)^{2} : 4 x + 4 x^{\frac{3}{2}} + x^{2}

= \int 4 x + 4 x^{\frac{3}{2}} + x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 x d x + \int 4 x^{\frac{3}{2}} d x + \int x^{2} d x

\int 4 x d x = 2 x^{2}

\int 4 x^{\frac{3}{2}} d x = \frac{8}{5} x^{\frac{5}{2}}

\int x^{2} d x = \frac{x ^{3}}{3}

= 2 x^{2} + \frac{8}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3}

Add a constant to the solution

= 2 x^{2} + \frac{8}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3} + C

What is the integral of x(2+sqrt(x))^2 ?
The integral of x(2+sqrt(x))^2 is 2x^2+8/5 x^{5/2}+(x^3)/3+C