What is the integral of (-x-8)/(x^2+9) ?

The integral of (-x-8)/(x^2+9) is -1/2 ln|x^2+9|-8/3 arctan(x/3)+C

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integral of (-x-8)/(x^2+9)

\int \frac{- x - 8}{x ^{2} + 9} d x

- \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{- x - 8}{x ^{2} + 9} d x

Expand

\frac{- x - 8}{x ^{2} + 9} : - \frac{x}{x ^{2} + 9} - \frac{8}{x ^{2} + 9}

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} + 9} d x - \int \frac{8}{x ^{2} + 9} d x

\int \frac{x}{x ^{2} + 9} d x = \frac{1}{2} ln x^{2} + 9

\int \frac{8}{x ^{2} + 9} d x = \frac{8}{3} arctan (\frac{x}{3})

= - \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3})

Add a constant to the solution

= - \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3}) + C

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What is the integral of (-x-8)/(x^2+9) ?
The integral of (-x-8)/(x^2+9) is -1/2 ln|x^2+9|-8/3 arctan(x/3)+C