What is the integral of (x-a)/(x-b) ?

The integral of (x-a)/(x-b) is bln|x-b|+x-aln|x-b|+C

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integral of (x-a)/(x-b)

\int \frac{x - a}{x - b} d x

b ln ∣ x - b ∣ + x - a ln ∣ x - b ∣ + C

Solution steps

\int \frac{x - a}{x - b} d x

Expand

\frac{x - a}{x - b} : \frac{x}{x - b} - \frac{a}{x - b}

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 - b} d x - \int \frac{a}{x - b} d x

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

\int \frac{a}{x - b} d x = a ln ∣ x - b ∣

= b ln ∣ x - b ∣ + x - a ln ∣ x - b ∣

Add a constant to the solution

= b ln ∣ x - b ∣ + x - a ln ∣ x - b ∣ + C

What is the integral of (x-a)/(x-b) ?
The integral of (x-a)/(x-b) is bln|x-b|+x-aln|x-b|+C