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

Popular >

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

\int \frac{( y + 3 )}{y ^{3} + 4 y ^{2}} d y

\int (x^{\frac{3}{2}} + 12 x + 2) d x

\int \frac{1}{1 - x + 3 1 - x} d x

\int \frac{6}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

\int \frac{6 x ^{3}}{x ^{4} + 8 x ^{2} + 16} d x

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