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

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

\int (x - 2)^{- \frac{1}{2}} d x

\frac{x}{( x - 2 ) ^{\frac{1}{2}}} + \frac{x - 4}{( x - 2 ) ^{\frac{1}{2}}} + C

Solution steps

\int (x - 2)^{- \frac{1}{2}} d x

Apply Integration By Parts

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

\int - \frac{x}{2 ( x - 2 ) ^{\frac{3}{2}}} d x = - \frac{x - 4}{( x - 2 ) ^{\frac{1}{2}}}

= \frac{x}{( x - 2 ) ^{\frac{1}{2}}} - (- \frac{x - 4}{( x - 2 ) ^{\frac{1}{2}}})

Simplify

= \frac{x}{( x - 2 ) ^{\frac{1}{2}}} + \frac{x - 4}{( x - 2 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{( x - 2 ) ^{\frac{1}{2}}} + \frac{x - 4}{( x - 2 ) ^{\frac{1}{2}}} + C

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