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

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

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

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

2 y ln ∣ - y + x ∣ + x + C

Solution steps

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

\frac{y + x}{- y + x} = \frac{2 y}{- y + x} + 1

= \int \frac{2 y}{- y + x} + 1 d x

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{2 y}{- y + x} d x + \int 1 d x

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

\int 1 d x = x

= 2 y ln ∣ - y + x ∣ + x

Add a constant to the solution

= 2 y ln ∣ - y + x ∣ + x + C

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