What is the integral of (3+2x)/(x^2-9) ?

The integral of (3+2x)/(x^2-9) is 1/2 ln|x+3|+3/2 ln|x-3|+C

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integral of (3+2x)/(x^2-9)

\int \frac{3 + 2 x}{x ^{2} - 9} d x

\frac{1}{2} ln ∣ x + 3 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{3 + 2 x}{x ^{2} - 9} d x

Take the partial fraction of

\frac{3 + 2 x}{x ^{2} - 9} : \frac{1}{2 ( x + 3 )} + \frac{3}{2 ( x - 3 )}

= \int \frac{1}{2 ( x + 3 )} + \frac{3}{2 ( 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{1}{2 ( x + 3 )} d x + \int \frac{3}{2 ( x - 3 )} d x

\int \frac{1}{2 ( x + 3 )} d x = \frac{1}{2} ln ∣ x + 3 ∣

\int \frac{3}{2 ( x - 3 )} d x = \frac{3}{2} ln ∣ x - 3 ∣

= \frac{1}{2} ln ∣ x + 3 ∣ + \frac{3}{2} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ x + 3 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ + C

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What is the integral of (3+2x)/(x^2-9) ?
The integral of (3+2x)/(x^2-9) is 1/2 ln|x+3|+3/2 ln|x-3|+C