What is the integral of (3x+8)/(x^2+2x) ?

The integral of (3x+8)/(x^2+2x) is 4ln|x|-ln|x+2|+C

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integral of (3x+8)/(x^2+2x)

\int \frac{3 x + 8}{x ^{2} + 2 x} d x

4 ln ∣ x ∣ - ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{3 x + 8}{x ^{2} + 2 x} d x

Take the partial fraction of

\frac{3 x + 8}{x ^{2} + 2 x} : \frac{4}{x} - \frac{1}{x + 2}

= \int \frac{4}{x} - \frac{1}{x + 2} 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{4}{x} d x - \int \frac{1}{x + 2} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

= 4 ln ∣ x ∣ - ln ∣ x + 2 ∣

Add a constant to the solution

= 4 ln ∣ x ∣ - ln ∣ x + 2 ∣ + C

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