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

The integral of (x-8)/(x^2+2) is 1/2 ln|x^2+2|-4sqrt(2)arctan(x/(sqrt(2)))+C

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

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

\frac{1}{2} ln x^{2} + 2 - 42 arctan (\frac{x}{2}) + C

Solution steps

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

Expand

\frac{x - 8}{x ^{2} + 2} : \frac{x}{x ^{2} + 2} - \frac{8}{x ^{2} + 2}

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

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

\int \frac{8}{x ^{2} + 2} d x = 42 arctan (\frac{x}{2})

= \frac{1}{2} ln x^{2} + 2 - 42 arctan (\frac{x}{2})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 2 - 42 arctan (\frac{x}{2}) + C

What is the integral of (x-8)/(x^2+2) ?
The integral of (x-8)/(x^2+2) is 1/2 ln|x^2+2|-4sqrt(2)arctan(x/(sqrt(2)))+C