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

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

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integral of (xy)^{1/2}

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

y^{\frac{1}{2}} \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

(y x)^{\frac{1}{2}} = y^{\frac{1}{2}} x^{\frac{1}{2}}

= \int y^{\frac{1}{2}} x^{\frac{1}{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= y^{\frac{1}{2}} \cdot \int x^{\frac{1}{2}} d x

Apply the Power Rule

: \frac{2}{3} x^{\frac{3}{2}}

= y^{\frac{1}{2}} \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= y^{\frac{1}{2}} \frac{2}{3} x^{\frac{3}{2}} + C

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