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

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

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

\int (ln (x - y) + \frac{x + y}{x - y}) d x

x ln (x - y) + 2 y ln ∣ x - y ∣ - y ln (x - y) + y + C

Solution steps

\int ln (x - y) + \frac{x + y}{x - y} 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{y + x}{- y + x} d x + \int ln (- y + x) d x

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

\int ln (- y + x) d x = - y ln (- y + x) + x ln (- y + x) + y - x

= 2 y ln ∣ - y + x ∣ + x - y ln (- y + x) + x ln (- y + x) + y - x

Simplify

= x ln (x - y) + 2 y ln ∣ x - y ∣ - y ln (x - y) + y

Add a constant to the solution

= x ln (x - y) + 2 y ln ∣ x - y ∣ - y ln (x - y) + y + C

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