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

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

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

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

2 ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

Solution steps

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

Expand

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

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

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

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

Simplify

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

= 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1}

Add a constant to the solution

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

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