What is the integral of (x-4)/(x^2+5x+6) ?

The integral of (x-4)/(x^2+5x+6) is -6ln|x+2|+7ln|x+3|+C

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integral of (x-4)/(x^2+5x+6)

\int \frac{x - 4}{x ^{2} + 5 x + 6} d x

- 6 ln ∣ x + 2 ∣ + 7 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x - 4}{x ^{2} + 5 x + 6} d x

Take the partial fraction of

\frac{x - 4}{x ^{2} + 5 x + 6} : - \frac{6}{x + 2} + \frac{7}{x + 3}

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

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

\int \frac{7}{x + 3} d x = 7 ln ∣ x + 3 ∣

= - 6 ln ∣ x + 2 ∣ + 7 ln ∣ x + 3 ∣

Add a constant to the solution

= - 6 ln ∣ x + 2 ∣ + 7 ln ∣ x + 3 ∣ + C

y = (\frac{cos ( x )}{1 - sin ( x )})^{2}

n = 1 \sum \infty \frac{1}{3 ^{n} + n ^{3}}

\int t^{- 2} e^{t} d t

(- 5, 2), (5, - 4)

\frac{d y}{d x} - \frac{x}{y} = 2 y

What is the integral of (x-4)/(x^2+5x+6) ?
The integral of (x-4)/(x^2+5x+6) is -6ln|x+2|+7ln|x+3|+C