What is the integral of (x-5)/((x-2)(x-3)^2) ?

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

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integral of (x-5)/((x-2)(x-3)^2)

\int \frac{x - 5}{( x - 2 ) ( x - 3 ) ^{2}} d x

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

Solution steps

\int \frac{x - 5}{( x - 2 ) ( x - 3 ) ^{2}} d x

Expand

\frac{x - 5}{( x - 2 ) ( x - 3 ) ^{2}} : \frac{x}{( x - 2 ) ( x - 3 ) ^{2}} - \frac{5}{( x - 2 ) ( x - 3 ) ^{2}}

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 - 2 ) ( x - 3 ) ^{2}} d x - \int \frac{5}{( x - 2 ) ( x - 3 ) ^{2}} d x

\int \frac{x}{( x - 2 ) ( x - 3 ) ^{2}} d x = - \frac{3}{x - 3} + 2 ln ∣ x - 2 ∣ - 2 ln ∣ x - 3 ∣

\int \frac{5}{( x - 2 ) ( x - 3 ) ^{2}} d x = 5 (ln ∣ x - 2 ∣ - ln ∣ x - 3 ∣ - \frac{1}{x - 3})

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

Simplify

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

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

Add a constant to the solution

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

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