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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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