What is the integral of (2x^2-8x-8)/((x-2)(x^2+4)) ?

The integral of (2x^2-8x-8)/((x-2)(x^2+4)) is -2ln|x-2|+2ln|x^2+4|+C

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integral of (2x^2-8x-8)/((x-2)(x^2+4))

\int \frac{2 x ^{2} - 8 x - 8}{( x - 2 ) ( x ^{2} + 4 )} d x

- 2 ln ∣ x - 2 ∣ + 2 ln x^{2} + 4 + C

Solution steps

\int \frac{2 x ^{2} - 8 x - 8}{( x - 2 ) ( x ^{2} + 4 )} d x

Take the partial fraction of

\frac{2 x ^{2} - 8 x - 8}{( x - 2 ) ( x ^{2} + 4 )} : - \frac{2}{x - 2} + \frac{4 x}{x ^{2} + 4}

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

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

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

= - 2 ln ∣ x - 2 ∣ + 2 ln x^{2} + 4

Add a constant to the solution

= - 2 ln ∣ x - 2 ∣ + 2 ln x^{2} + 4 + C

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What is the integral of (2x^2-8x-8)/((x-2)(x^2+4)) ?
The integral of (2x^2-8x-8)/((x-2)(x^2+4)) is -2ln|x-2|+2ln|x^2+4|+C