What is the integral of (8x+28)/(x^2+8x+12) ?

The integral of (8x+28)/(x^2+8x+12) is 3ln|x+2|+5ln|x+6|+C

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integral of (8x+28)/(x^2+8x+12)

\int \frac{8 x + 28}{x ^{2} + 8 x + 12} d x

3 ln ∣ x + 2 ∣ + 5 ln ∣ x + 6 ∣ + C

Solution steps

\int \frac{8 x + 28}{x ^{2} + 8 x + 12} d x

Take the partial fraction of

\frac{8 x + 28}{x ^{2} + 8 x + 12} : \frac{3}{x + 2} + \frac{5}{x + 6}

= \int \frac{3}{x + 2} + \frac{5}{x + 6} 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{5}{x + 6} d x

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

\int \frac{5}{x + 6} d x = 5 ln ∣ x + 6 ∣

= 3 ln ∣ x + 2 ∣ + 5 ln ∣ x + 6 ∣

Add a constant to the solution

= 3 ln ∣ x + 2 ∣ + 5 ln ∣ x + 6 ∣ + C

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(4, 2), (- 16, - 10)

t^{- \frac{1}{2}}

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What is the integral of (8x+28)/(x^2+8x+12) ?
The integral of (8x+28)/(x^2+8x+12) is 3ln|x+2|+5ln|x+6|+C