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

The integral of (4-x)^{-1/2} is x/((4-x)^{1/2)}+(x-8)/((4-x)^{1/2)}+C

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

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

\frac{x}{( 4 - x ) ^{\frac{1}{2}}} + \frac{x - 8}{( 4 - x ) ^{\frac{1}{2}}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x}{2 ( 4 - x ) ^{\frac{3}{2}}} d x = - \frac{x - 8}{( 4 - x ) ^{\frac{1}{2}}}

= \frac{x}{( 4 - x ) ^{\frac{1}{2}}} - (- \frac{x - 8}{( 4 - x ) ^{\frac{1}{2}}})

Simplify

= \frac{x}{( 4 - x ) ^{\frac{1}{2}}} + \frac{x - 8}{( 4 - x ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{( 4 - x ) ^{\frac{1}{2}}} + \frac{x - 8}{( 4 - x ) ^{\frac{1}{2}}} + C

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What is the integral of (4-x)^{-1/2} ?
The integral of (4-x)^{-1/2} is x/((4-x)^{1/2)}+(x-8)/((4-x)^{1/2)}+C