What is the integral of (x-1)/((x-2)(x+3)) ?

The integral of (x-1)/((x-2)(x+3)) is 1/5 ln|x-2|+4/5 ln|x+3|+C

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integral of (x-1)/((x-2)(x+3))

\int \frac{x - 1}{( x - 2 ) ( x + 3 )} d x

\frac{1}{5} ln ∣ x - 2 ∣ + \frac{4}{5} ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x - 1}{( x - 2 ) ( x + 3 )} d x

Take the partial fraction of

\frac{x - 1}{( x - 2 ) ( x + 3 )} : \frac{1}{5 ( x - 2 )} + \frac{4}{5 ( x + 3 )}

= \int \frac{1}{5 ( x - 2 )} + \frac{4}{5 ( x + 3 )} 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}{5 ( x - 2 )} d x + \int \frac{4}{5 ( x + 3 )} d x

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

\int \frac{4}{5 ( x + 3 )} d x = \frac{4}{5} ln ∣ x + 3 ∣

= \frac{1}{5} ln ∣ x - 2 ∣ + \frac{4}{5} ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ x - 2 ∣ + \frac{4}{5} ln ∣ x + 3 ∣ + C

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What is the integral of (x-1)/((x-2)(x+3)) ?
The integral of (x-1)/((x-2)(x+3)) is 1/5 ln|x-2|+4/5 ln|x+3|+C