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

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

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

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

ln ∣ y + x ∣ + \frac{y}{y + x} - \frac{2 y}{y + x} + C

Solution steps

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

Expand

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

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

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

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

= ln ∣ y + x ∣ + \frac{y}{y + x} - \frac{2 y}{y + x}

Add a constant to the solution

= ln ∣ y + x ∣ + \frac{y}{y + x} - \frac{2 y}{y + x} + C

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