What is the integral of (x+2)/(x^{1/2)} ?

The integral of (x+2)/(x^{1/2)} is 2/3 x^{3/2}+4x^{1/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x+2)/(x^{1/2)}

\int \frac{x + 2}{x ^{\frac{1}{2}}} d x

\frac{2}{3} x^{\frac{3}{2}} + 4 x^{\frac{1}{2}} + C

Solution steps

\int \frac{x + 2}{x ^{\frac{1}{2}}} d x

Expand

\frac{x + 2}{x ^{\frac{1}{2}}} : x^{\frac{1}{2}} + \frac{2}{x ^{\frac{1}{2}}}

= \int x^{\frac{1}{2}} + \frac{2}{x ^{\frac{1}{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 x^{\frac{1}{2}} d x + \int \frac{2}{x ^{\frac{1}{2}}} d x

\int x^{\frac{1}{2}} d x = \frac{2}{3} x^{\frac{3}{2}}

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

= \frac{2}{3} x^{\frac{3}{2}} + 4 x^{\frac{1}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 4 x^{\frac{1}{2}} + C

What is the integral of (x+2)/(x^{1/2)} ?
The integral of (x+2)/(x^{1/2)} is 2/3 x^{3/2}+4x^{1/2}+C