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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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