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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

- ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + x - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1 : x - \frac{3}{2} ln \frac{x}{2} + 1 + \frac{3}{2} ln \frac{x}{2} - 1

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

Add a constant to the solution

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

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