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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int - 2 d x = - 2 x

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

Simplify

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

Add a constant to the solution

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

y^{^{''}} - 14 y^{^{'}} + 51 y = 0

x y^{^{'}} + y = - 8 x y^{2}, y (1) = 1

\int \frac{7 x}{x ^{2} + 4} d x

(3 cos (x))^{^{'}}

\frac{\partial}{\partial x} (x^{3} y^{2} - 2 x e^{x y})

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