What is the integral of (5x-3)/((x+1)(x-3)) ?

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

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integral of (5x-3)/((x+1)(x-3))

\int \frac{5 x - 3}{( x + 1 ) ( x - 3 )} d x

2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{5 x - 3}{( x + 1 ) ( x - 3 )} d x

Apply u-substitution

= \int \frac{5 u - 8}{u ( u - 4 )} d u

Take the partial fraction of

\frac{5 u - 8}{u ( u - 4 )} : \frac{2}{u} + \frac{3}{u - 4}

= \int \frac{2}{u} + \frac{3}{u - 4} 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{2}{u} d u + \int \frac{3}{u - 4} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{3}{u - 4} d u = 3 ln ∣ u - 4 ∣

= 2 ln ∣ u ∣ + 3 ln ∣ u - 4 ∣

Substitute back

u = x + 1

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x + 1 - 4 ∣

Simplify

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

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