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

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

integral of (x-y)/(x+y)

\int \frac{x - y}{x + y} d y

2 x ln ∣ x + y ∣ - y + C

Solution steps

\int \frac{x - y}{x + y} d y

Expand

\frac{x - y}{x + y} : \frac{x}{x + y} - \frac{y}{x + y}

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}{x + y} d y - \int \frac{y}{x + y} d y

\int \frac{x}{x + y} d y = x ln ∣ x + y ∣

\int \frac{y}{x + y} d y = - x ln ∣ x + y ∣ + y

= x ln ∣ x + y ∣ - (- x ln ∣ x + y ∣ + y)

Simplify

= 2 x ln ∣ x + y ∣ - y

Add a constant to the solution

= 2 x ln ∣ x + y ∣ - y + C

\int 9 x^{\frac{1}{2}} d x

x \to \infty lim (2 x - 3)

t a y l or \frac{5}{x}, 4

n = 0 \sum \infty \frac{1}{e ^{5 n}}

\int (- 4 x^{3} + 2 e^{- 2 x} + \frac{1}{2 x}) d x

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