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

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

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

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

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

Solution steps

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

\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

\int x d x = \frac{x ^{2}}{2}

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

Add a constant to the solution

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

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