What is the integral of (x+2)/(x^3+2x^2+5x) ?

The integral of (x+2)/(x^3+2x^2+5x) is 2/5 ln|x|+1/10 (-2ln|x^2+2x+5|+3arctan(1/2 (x+1)))+C

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integral of (x+2)/(x^3+2x^2+5x)

\int \frac{x + 2}{x ^{3} + 2 x ^{2} + 5 x} d x

\frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1))) + C

Solution steps

\int \frac{x + 2}{x ^{3} + 2 x ^{2} + 5 x} d x

Take the partial fraction of

\frac{x + 2}{x ^{3} + 2 x ^{2} + 5 x} : \frac{2}{5 x} + \frac{- 2 x + 1}{5 ( x ^{2} + 2 x + 5 )}

= \int \frac{2}{5 x} + \frac{- 2 x + 1}{5 ( x ^{2} + 2 x + 5 )} 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}{5 x} d x + \int \frac{- 2 x + 1}{5 ( x ^{2} + 2 x + 5 )} d x

\int \frac{2}{5 x} d x = \frac{2}{5} ln ∣ x ∣

\int \frac{- 2 x + 1}{5 ( x ^{2} + 2 x + 5 )} d x = \frac{1}{5} (- ln x^{2} + 2 x + 5 + arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} arctan (\frac{1}{2} (x + 1)))

= \frac{2}{5} ln ∣ x ∣ + \frac{1}{5} (- ln x^{2} + 2 x + 5 + arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} arctan (\frac{1}{2} (x + 1)))

Simplify

\frac{2}{5} ln ∣ x ∣ + \frac{1}{5} (- ln x^{2} + 2 x + 5 + arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} arctan (\frac{1}{2} (x + 1))) : \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1)))

= \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1)))

Add a constant to the solution

= \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1))) + C

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What is the integral of (x+2)/(x^3+2x^2+5x) ?
The integral of (x+2)/(x^3+2x^2+5x) is 2/5 ln|x|+1/10 (-2ln|x^2+2x+5|+3arctan(1/2 (x+1)))+C