What is the integral of (3x+1)/(x^2-x-6) ?

The integral of (3x+1)/(x^2-x-6) is ln|x+2|+2ln|x-3|+C

Popular >

integral of (3x+1)/(x^2-x-6)

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

ln ∣ x + 2 ∣ + 2 ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

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

= ln ∣ x + 2 ∣ + 2 ln ∣ x - 3 ∣

Add a constant to the solution

= ln ∣ x + 2 ∣ + 2 ln ∣ x - 3 ∣ + C

\int e^{- \frac{3}{5} x} d x

\int 12 x^{5} y^{2} d x

\int \frac{1}{x 1 - 2 x ^{2}} d x

\int + 3 x^{5} x^{3} + 1 d x

\int \frac{1}{tan ( \frac{x}{5} )} d x

What is the integral of (3x+1)/(x^2-x-6) ?
The integral of (3x+1)/(x^2-x-6) is ln|x+2|+2ln|x-3|+C