What is the integral of (x^2-2x+6)/((x^2+2)(x-2)) ?

The integral of (x^2-2x+6)/((x^2+2)(x-2)) is -sqrt(2)arctan(x/(sqrt(2)))+ln|x-2|+C

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integral of (x^2-2x+6)/((x^2+2)(x-2))

\int \frac{x ^{2} - 2 x + 6}{( x ^{2} + 2 ) ( x - 2 )} d x

- 2 arctan (\frac{x}{2}) + ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{2} - 2 x + 6}{( x ^{2} + 2 ) ( x - 2 )} d x

Take the partial fraction of

\frac{x ^{2} - 2 x + 6}{( x ^{2} + 2 ) ( x - 2 )} : - \frac{2}{x ^{2} + 2} + \frac{1}{x - 2}

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

\int \frac{2}{x ^{2} + 2} d x = 2 arctan (\frac{x}{2})

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

= - 2 arctan (\frac{x}{2}) + ln ∣ x - 2 ∣

Add a constant to the solution

= - 2 arctan (\frac{x}{2}) + ln ∣ x - 2 ∣ + C

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What is the integral of (x^2-2x+6)/((x^2+2)(x-2)) ?
The integral of (x^2-2x+6)/((x^2+2)(x-2)) is -sqrt(2)arctan(x/(sqrt(2)))+ln|x-2|+C