What is the integral of (21x-69)/((x-2)(x-5)) ?

The integral of (21x-69)/((x-2)(x-5)) is 9ln|x-2|+12ln|x-5|+C

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integral of (21x-69)/((x-2)(x-5))

\int \frac{21 x - 69}{( x - 2 ) ( x - 5 )} d x

9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{21 x - 69}{( x - 2 ) ( x - 5 )} d x

Apply u-substitution

= \int \frac{21 u - 27}{u ( u - 3 )} d u

Take the partial fraction of

\frac{21 u - 27}{u ( u - 3 )} : \frac{9}{u} + \frac{12}{u - 3}

= \int \frac{9}{u} + \frac{12}{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{9}{u} d u + \int \frac{12}{u - 3} d u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

\int \frac{12}{u - 3} d u = 12 ln ∣ u - 3 ∣

= 9 ln ∣ u ∣ + 12 ln ∣ u - 3 ∣

Substitute back

u = x - 2

= 9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 2 - 3 ∣

Simplify

= 9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 5 ∣

Add a constant to the solution

= 9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 5 ∣ + C

What is the integral of (21x-69)/((x-2)(x-5)) ?
The integral of (21x-69)/((x-2)(x-5)) is 9ln|x-2|+12ln|x-5|+C