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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

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