What is the integral of (20x-15)/((2x-3)(x+1)) ?

The integral of (20x-15)/((2x-3)(x+1)) is 3ln|2x-3|+7ln|x+1|+C

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integral of (20x-15)/((2x-3)(x+1))

\int \frac{20 x - 15}{( 2 x - 3 ) ( x + 1 )} d x

3 ln ∣ 2 x - 3 ∣ + 7 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{20 x - 15}{( 2 x - 3 ) ( x + 1 )} d x

Take the partial fraction of

\frac{20 x - 15}{( 2 x - 3 ) ( x + 1 )} : \frac{6}{2 x - 3} + \frac{7}{x + 1}

= \int \frac{6}{2 x - 3} + \frac{7}{x + 1} d x

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{6}{2 x - 3} d x + \int \frac{7}{x + 1} d x

\int \frac{6}{2 x - 3} d x = 3 ln ∣ 2 x - 3 ∣

\int \frac{7}{x + 1} d x = 7 ln ∣ x + 1 ∣

= 3 ln ∣ 2 x - 3 ∣ + 7 ln ∣ x + 1 ∣

Add a constant to the solution

= 3 ln ∣ 2 x - 3 ∣ + 7 ln ∣ x + 1 ∣ + C

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What is the integral of (20x-15)/((2x-3)(x+1)) ?
The integral of (20x-15)/((2x-3)(x+1)) is 3ln|2x-3|+7ln|x+1|+C