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

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

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

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

2 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + C

Solution steps

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

Expand

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

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

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

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

Simplify

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

= 2 ln ∣ x - 1 ∣ - \frac{5}{x - 1}

Add a constant to the solution

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

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