What is the integral from a to b of (x+y)/(x-y) ?

The integral from a to b of (x+y)/(x-y) is 2y(ln|-y+b|-ln|-y+a|)+b-a

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integral from a to b of (x+y)/(x-y)

\int_{a}^{b} \frac{x + y}{x - y} d x

2 y (ln ∣ - y + b ∣ - ln ∣ - y + a ∣) + b - a

Solution steps

\int_{a}^{b} \frac{x + y}{x - y} d x

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

= \int_{a}^{b} \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_{a}^{b} \frac{2 y}{- y + x} d x + \int_{a}^{b} 1 d x

\int_{a}^{b} \frac{2 y}{- y + x} d x = 2 y (ln ∣ - y + b ∣ - ln ∣ - y + a ∣)

\int_{a}^{b} 1 d x = b - a

= 2 y (ln ∣ - y + b ∣ - ln ∣ - y + a ∣) + b - a

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What is the integral from a to b of (x+y)/(x-y) ?
The integral from a to b of (x+y)/(x-y) is 2y(ln|-y+b|-ln|-y+a|)+b-a