What is the integral of (6x+4)/(x^2+8x+15) ?

The integral of (6x+4)/(x^2+8x+15) is -7ln|x+3|+13ln|x+5|+C

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integral of (6x+4)/(x^2+8x+15)

\int \frac{6 x + 4}{x ^{2} + 8 x + 15} d x

- 7 ln ∣ x + 3 ∣ + 13 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{6 x + 4}{x ^{2} + 8 x + 15} d x

Take the partial fraction of

\frac{6 x + 4}{x ^{2} + 8 x + 15} : - \frac{7}{x + 3} + \frac{13}{x + 5}

= \int - \frac{7}{x + 3} + \frac{13}{x + 5} 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{7}{x + 3} d x + \int \frac{13}{x + 5} d x

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

\int \frac{13}{x + 5} d x = 13 ln ∣ x + 5 ∣

= - 7 ln ∣ x + 3 ∣ + 13 ln ∣ x + 5 ∣

Add a constant to the solution

= - 7 ln ∣ x + 3 ∣ + 13 ln ∣ x + 5 ∣ + C

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What is the integral of (6x+4)/(x^2+8x+15) ?
The integral of (6x+4)/(x^2+8x+15) is -7ln|x+3|+13ln|x+5|+C