What is the integral of (14x-11)/((2x-3)(x+1)) ?

The integral of (14x-11)/((2x-3)(x+1)) is 2ln|2x-3|+5ln|x+1|+C

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integral of (14x-11)/((2x-3)(x+1))

\int \frac{14 x - 11}{( 2 x - 3 ) ( x + 1 )} d x

2 ln ∣ 2 x - 3 ∣ + 5 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{14 x - 11}{( 2 x - 3 ) ( x + 1 )} d x

Take the partial fraction of

\frac{14 x - 11}{( 2 x - 3 ) ( x + 1 )} : \frac{4}{2 x - 3} + \frac{5}{x + 1}

= \int \frac{4}{2 x - 3} + \frac{5}{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{4}{2 x - 3} d x + \int \frac{5}{x + 1} d x

\int \frac{4}{2 x - 3} d x = 2 ln ∣ 2 x - 3 ∣

\int \frac{5}{x + 1} d x = 5 ln ∣ x + 1 ∣

= 2 ln ∣ 2 x - 3 ∣ + 5 ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ 2 x - 3 ∣ + 5 ln ∣ x + 1 ∣ + C

What is the integral of (14x-11)/((2x-3)(x+1)) ?
The integral of (14x-11)/((2x-3)(x+1)) is 2ln|2x-3|+5ln|x+1|+C