What is the integral of (-8x+162)/(x^2+9) ?

The integral of (-8x+162)/(x^2+9) is -4ln|x^2+9|+54arctan(x/3)+C

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integral of (-8x+162)/(x^2+9)

\int \frac{- 8 x + 162}{x ^{2} + 9} d x

- 4 ln x^{2} + 9 + 54 arctan (\frac{x}{3}) + C

Solution steps

\int \frac{- 8 x + 162}{x ^{2} + 9} d x

Expand

\frac{- 8 x + 162}{x ^{2} + 9} : - \frac{8 x}{x ^{2} + 9} + \frac{162}{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{8 x}{x ^{2} + 9} d x + \int \frac{162}{x ^{2} + 9} d x

\int \frac{8 x}{x ^{2} + 9} d x = 4 ln x^{2} + 9

\int \frac{162}{x ^{2} + 9} d x = 54 arctan (\frac{x}{3})

= - 4 ln x^{2} + 9 + 54 arctan (\frac{x}{3})

Add a constant to the solution

= - 4 ln x^{2} + 9 + 54 arctan (\frac{x}{3}) + C

What is the integral of (-8x+162)/(x^2+9) ?
The integral of (-8x+162)/(x^2+9) is -4ln|x^2+9|+54arctan(x/3)+C