What is the integral of (x+3)/(x^2-4) ?

The integral of (x+3)/(x^2-4) is 1/2 ln|x^2-4|-3/4 (ln| x/2+1|-ln| x/2-1|)+C

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integral of (x+3)/(x^2-4)

\int \frac{x + 3}{x ^{2} - 4} d x

\frac{1}{2} ln x^{2} - 4 - \frac{3}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1) + C

Solution steps

\int \frac{x + 3}{x ^{2} - 4} d x

Expand

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

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{x}{x ^{2} - 4} d x + \int \frac{3}{x ^{2} - 4} d x

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

\int \frac{3}{x ^{2} - 4} d x = - \frac{3}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1)

= \frac{1}{2} ln x^{2} - 4 - \frac{3}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1)

Add a constant to the solution

= \frac{1}{2} ln x^{2} - 4 - \frac{3}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1) + C

What is the integral of (x+3)/(x^2-4) ?
The integral of (x+3)/(x^2-4) is 1/2 ln|x^2-4|-3/4 (ln| x/2+1|-ln| x/2-1|)+C