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

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

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

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

2 (x - 1) + 5 ln ∣ x - 1 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u + 5}{u} d u

= \int \frac{2 u + 5}{u} d u

Expand

\frac{2 u + 5}{u} : 2 + \frac{5}{u}

= \int 2 + \frac{5}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 d u + \int \frac{5}{u} d u

\int 2 d u = 2 u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= 2 u + 5 ln ∣ u ∣

Substitute back

u = x - 1

= 2 (x - 1) + 5 ln ∣ x - 1 ∣

Add a constant to the solution

= 2 (x - 1) + 5 ln ∣ x - 1 ∣ + C

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