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

The integral of y/((x+y)^2) is ln|x+y|+x/(x+y)+C

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

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

ln ∣ x + y ∣ + \frac{x}{x + y} + C

Solution steps

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

Apply u-substitution

: \int \frac{u - x}{u ^{2}} d u

= \int \frac{u - x}{u ^{2}} d u

Expand

\frac{u - x}{u ^{2}} : \frac{1}{u} - \frac{x}{u ^{2}}

= \int \frac{1}{u} - \frac{x}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u - \int \frac{x}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{x}{u ^{2}} d u = - \frac{x}{u}

= ln ∣ u ∣ - (- \frac{x}{u})

Substitute back

u = x + y

= ln ∣ x + y ∣ - (- \frac{x}{x + y})

Simplify

= ln ∣ x + y ∣ + \frac{x}{x + y}

Add a constant to the solution

= ln ∣ x + y ∣ + \frac{x}{x + y} + C

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