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

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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

= \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

\int (x - sin (x) + cos (x)) d x

\int \frac{1}{( a - x )} d x

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

\int \frac{t}{4 - t ^{4}} d t

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