What is the integral of ((x-y))/((x+y)^3) ?

The integral of ((x-y))/((x+y)^3) is (y-x)/(2(y+x)^2)-1/(2(y+x))+C

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integral of ((x-y))/((x+y)^3)

integral

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

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

Solution steps

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

Factor

- y + x : - (y - x)

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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