What is the integral of (3x+16)/(x^2-x-6) ?

The integral of (3x+16)/(x^2-x-6) is -2ln|x+2|+5ln|x-3|+C

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integral of (3x+16)/(x^2-x-6)

\int \frac{3 x + 16}{x ^{2} - x - 6} d x

- 2 ln ∣ x + 2 ∣ + 5 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{3 x + 16}{x ^{2} - x - 6} d x

Take the partial fraction of

\frac{3 x + 16}{x ^{2} - x - 6} : - \frac{2}{x + 2} + \frac{5}{x - 3}

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

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

\int \frac{5}{x - 3} d x = 5 ln ∣ x - 3 ∣

= - 2 ln ∣ x + 2 ∣ + 5 ln ∣ x - 3 ∣

Add a constant to the solution

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

x \to 1 lim (0.8 x + 1.2)

x \to 0 lim (cos (x) sin (x))

y^{^{'}} = 2 y + 3

\frac{d x}{d t} = (a - b \cdot t) x

\frac{d}{d x} ((x^{2} + x)^{2})

What is the integral of (3x+16)/(x^2-x-6) ?
The integral of (3x+16)/(x^2-x-6) is -2ln|x+2|+5ln|x-3|+C