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

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

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integral of (x+y)^{-2}(x^2+2xy-y^2)

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

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

Solution steps

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

Simplify

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

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

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

v^{2} d x + x (x + v) d v = 0

\int \frac{x ^{3} + 2 x + 5}{x ^{2}} d x

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