What is the integral of (2xy(y+2))/((2x^2+y^2)^2) ?

The integral of (2xy(y+2))/((2x^2+y^2)^2) is -(y(y+2))/(2(2x^2+y^2))+C

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integral of (2xy(y+2))/((2x^2+y^2)^2)

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

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

Solution steps

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

Take the constant out:

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

= (2 + y) \cdot 2 y \cdot \int \frac{x}{( y ^{2} + 2 x ^{2} ) ^{2}} d x

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

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

Simplify

(2 + y) \cdot 2 y (- \frac{1}{4 ( y ^{2} + 2 x ^{2} )}) : - \frac{y ( y + 2 )}{2 ( 2 x ^{2} + y ^{2} )}

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

Add a constant to the solution

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

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What is the integral of (2xy(y+2))/((2x^2+y^2)^2) ?
The integral of (2xy(y+2))/((2x^2+y^2)^2) is -(y(y+2))/(2(2x^2+y^2))+C