What is the integral of+sqrt(xy-y^2) ?

The integral of+sqrt(xy-y^2) is (2(-y^2+yx)^{3/2})/(3y)+C

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integral of+sqrt(xy-y^2)

\int + x y - y^{2} d x

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

Solution steps

\int x y - y^{2} d x

Apply u-substitution

: \int \frac{u}{y} d u

= \int \frac{u}{y} d u

Take the constant out:

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

= \frac{1}{y} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{y} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = - y^{2} + y x

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

Simplify

\frac{1}{y} \cdot \frac{2}{3} (- y^{2} + y x)^{\frac{3}{2}} : \frac{2 ( - y ^{2} + y x ) ^{\frac{3}{2}}}{3 y}

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

Add a constant to the solution

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

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What is the integral of+sqrt(xy-y^2) ?
The integral of+sqrt(xy-y^2) is (2(-y^2+yx)^{3/2})/(3y)+C