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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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