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

The integral of (((2+sqrt(x))^2)/(sqrt(x))) is 8sqrt(x)+4x+2/3 x^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

8 x + 4 x + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Expand

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

= \int \frac{4}{x} + 4 + x 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 \frac{4}{x} d x + \int 4 d x + \int x d x

\int \frac{4}{x} d x = 8 x

\int 4 d x = 4 x

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

= 8 x + 4 x + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= 8 x + 4 x + \frac{2}{3} x^{\frac{3}{2}} + C

What is the integral of (((2+sqrt(x))^2)/(sqrt(x))) ?
The integral of (((2+sqrt(x))^2)/(sqrt(x))) is 8sqrt(x)+4x+2/3 x^{3/2}+C