What is the integral of (-2x^2+3x-15)/((x^2+9)(x+1)) ?

The integral of (-2x^2+3x-15)/((x^2+9)(x+1)) is arctan(x/3)-2ln|x+1|+C

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integral of (-2x^2+3x-15)/((x^2+9)(x+1))

\int \frac{- 2 x ^{2} + 3 x - 15}{( x ^{2} + 9 ) ( x + 1 )} d x

arctan (\frac{x}{3}) - 2 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{- 2 x ^{2} + 3 x - 15}{( x ^{2} + 9 ) ( x + 1 )} d x

Take the partial fraction of

\frac{- 2 x ^{2} + 3 x - 15}{( x ^{2} + 9 ) ( x + 1 )} : \frac{3}{x ^{2} + 9} - \frac{2}{x + 1}

= \int \frac{3}{x ^{2} + 9} - \frac{2}{x + 1} 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{3}{x ^{2} + 9} d x - \int \frac{2}{x + 1} d x

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

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

= arctan (\frac{x}{3}) - 2 ln ∣ x + 1 ∣

Add a constant to the solution

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

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What is the integral of (-2x^2+3x-15)/((x^2+9)(x+1)) ?
The integral of (-2x^2+3x-15)/((x^2+9)(x+1)) is arctan(x/3)-2ln|x+1|+C