What is the integral of x/((x-1)(2-x)) ?

The integral of x/((x-1)(2-x)) is ln|x-1|-2ln|x-2|+C

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integral of x/((x-1)(2-x))

\int \frac{x}{( x - 1 ) ( 2 - x )} d x

ln ∣ x - 1 ∣ - 2 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x}{( x - 1 ) ( 2 - x )} d x

Take the partial fraction of

\frac{x}{( x - 1 ) ( 2 - x )} : \frac{1}{x - 1} - \frac{2}{x - 2}

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

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

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

= ln ∣ x - 1 ∣ - 2 ln ∣ x - 2 ∣

Add a constant to the solution

= ln ∣ x - 1 ∣ - 2 ln ∣ x - 2 ∣ + C

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What is the integral of x/((x-1)(2-x)) ?
The integral of x/((x-1)(2-x)) is ln|x-1|-2ln|x-2|+C