What is the integral of 1/(x^3+4x^2+5x) ?

The integral of 1/(x^3+4x^2+5x) is 1/5 ln|x|+1/10 (-ln|x^2+4x+5|-4arctan(x+2))+C

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integral of 1/(x^3+4x^2+5x)

\int \frac{1}{x ^{3} + 4 x ^{2} + 5 x} d x

\frac{1}{5} ln ∣ x ∣ + \frac{1}{10} (- ln x^{2} + 4 x + 5 - 4 arctan (x + 2)) + C

Solution steps

\int \frac{1}{x ^{3} + 4 x ^{2} + 5 x} d x

Take the partial fraction of

\frac{1}{x ^{3} + 4 x ^{2} + 5 x} : \frac{1}{5 x} + \frac{- x - 4}{5 ( x ^{2} + 4 x + 5 )}

= \int \frac{1}{5 x} + \frac{- x - 4}{5 ( x ^{2} + 4 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{1}{5 x} d x + \int \frac{- x - 4}{5 ( x ^{2} + 4 x + 5 )} d x

\int \frac{1}{5 x} d x = \frac{1}{5} ln ∣ x ∣

\int \frac{- x - 4}{5 ( x ^{2} + 4 x + 5 )} d x = \frac{1}{10} (- ln x^{2} + 4 x + 5 - 4 arctan (x + 2))

= \frac{1}{5} ln ∣ x ∣ + \frac{1}{10} (- ln x^{2} + 4 x + 5 - 4 arctan (x + 2))

Add a constant to the solution

= \frac{1}{5} ln ∣ x ∣ + \frac{1}{10} (- ln x^{2} + 4 x + 5 - 4 arctan (x + 2)) + C

What is the integral of 1/(x^3+4x^2+5x) ?
The integral of 1/(x^3+4x^2+5x) is 1/5 ln|x|+1/10 (-ln|x^2+4x+5|-4arctan(x+2))+C