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

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

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

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

2 (x - 2) + 3 ln ∣ x - 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u + 3}{u} d u

= \int \frac{2 u + 3}{u} d u

Expand

\frac{2 u + 3}{u} : 2 + \frac{3}{u}

= \int 2 + \frac{3}{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{3}{u} d u

\int 2 d u = 2 u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= 2 u + 3 ln ∣ u ∣

Substitute back

u = x - 2

= 2 (x - 2) + 3 ln ∣ x - 2 ∣

Add a constant to the solution

= 2 (x - 2) + 3 ln ∣ x - 2 ∣ + C

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