What is the integral of (x^2+1)/(x^2-3) ?

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

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integral of (x^2+1)/(x^2-3)

\int \frac{x ^{2} + 1}{x ^{2} - 3} d x

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

Solution steps

\int \frac{x ^{2} + 1}{x ^{2} - 3} d x

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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