What is the integral of (2x+11)/(x^2+x-6) ?

The integral of (2x+11)/(x^2+x-6) is 3ln|x-2|-ln|x+3|+C

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integral of (2x+11)/(x^2+x-6)

\int \frac{2 x + 11}{x ^{2} + x - 6} d x

3 ln ∣ x - 2 ∣ - ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{2 x + 11}{x ^{2} + x - 6} d x

Take the partial fraction of

\frac{2 x + 11}{x ^{2} + x - 6} : \frac{3}{x - 2} - \frac{1}{x + 3}

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

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

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

= 3 ln ∣ x - 2 ∣ - ln ∣ x + 3 ∣

Add a constant to the solution

= 3 ln ∣ x - 2 ∣ - ln ∣ x + 3 ∣ + C

What is the integral of (2x+11)/(x^2+x-6) ?
The integral of (2x+11)/(x^2+x-6) is 3ln|x-2|-ln|x+3|+C