What is the integral of (5x+1)/((x-1)(x+2)) ?

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

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integral of (5x+1)/((x-1)(x+2))

\int \frac{5 x + 1}{( x - 1 ) ( x + 2 )} d x

3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{5 x + 1}{( x - 1 ) ( x + 2 )} d x

Apply u-substitution

: \int \frac{5 u - 9}{u ( u - 3 )} d u

= \int \frac{5 u - 9}{u ( u - 3 )} d u

Take the partial fraction of

\frac{5 u - 9}{u ( u - 3 )} : \frac{3}{u} + \frac{2}{u - 3}

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

\int \frac{3}{u} d u = 3 ln ∣ u ∣

\int \frac{2}{u - 3} d u = 2 ln ∣ u - 3 ∣

= 3 ln ∣ u ∣ + 2 ln ∣ u - 3 ∣

Substitute back

u = x + 2

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x + 2 - 3 ∣

Simplify

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣

Add a constant to the solution

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

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