What is the integral of (3-x)^{(-1)/2} ?

The integral of (3-x)^{(-1)/2} is x/((3-x)^{1/2)}+(x-6)/((3-x)^{1/2)}+C

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integral of (3-x)^{(-1)/2}

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

\frac{x}{( 3 - x ) ^{\frac{1}{2}}} + \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}} + C

Solution steps

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

\frac{- 1}{2} = - \frac{1}{2}

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

Apply Integration By Parts

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

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

\int \frac{x}{2 ( 3 - x ) ^{\frac{3}{2}}} d x = - \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}}

= \frac{x}{( 3 - x ) ^{\frac{1}{2}}} - (- \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}})

Simplify

= \frac{x}{( 3 - x ) ^{\frac{1}{2}}} + \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{( 3 - x ) ^{\frac{1}{2}}} + \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}} + C

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What is the integral of (3-x)^{(-1)/2} ?
The integral of (3-x)^{(-1)/2} is x/((3-x)^{1/2)}+(x-6)/((3-x)^{1/2)}+C