What is the integral of (5x-13)/((x-3)(x-2)) ?

The integral of (5x-13)/((x-3)(x-2)) is 2ln|x-3|+3ln|x-2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (5x-13)/((x-3)(x-2))

\int \frac{5 x - 13}{( x - 3 ) ( x - 2 )} d x

2 ln ∣ x - 3 ∣ + 3 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{5 x - 13}{( x - 3 ) ( x - 2 )} d x

Apply u-substitution

= \int \frac{5 u + 2}{u ( u + 1 )} d u

Take the partial fraction of

\frac{5 u + 2}{u ( u + 1 )} : \frac{2}{u} + \frac{3}{u + 1}

= \int \frac{2}{u} + \frac{3}{u + 1} 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 + 1} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{3}{u + 1} d u = 3 ln ∣ u + 1 ∣

= 2 ln ∣ u ∣ + 3 ln ∣ u + 1 ∣

Substitute back

u = x - 3

= 2 ln ∣ x - 3 ∣ + 3 ln ∣ x - 3 + 1 ∣

Simplify

= 2 ln ∣ x - 3 ∣ + 3 ln ∣ x - 2 ∣

Add a constant to the solution

= 2 ln ∣ x - 3 ∣ + 3 ln ∣ x - 2 ∣ + C

What is the integral of (5x-13)/((x-3)(x-2)) ?
The integral of (5x-13)/((x-3)(x-2)) is 2ln|x-3|+3ln|x-2|+C