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

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

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

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

ln ∣ x ∣ - \frac{1}{2} ln 2 x + 1 - \frac{1}{2} ln 2 x - 1 + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= ln ∣ x ∣ - \frac{1}{2} ln 2 x + 1 - \frac{1}{2} ln 2 x - 1

Add a constant to the solution

= ln ∣ x ∣ - \frac{1}{2} ln 2 x + 1 - \frac{1}{2} ln 2 x - 1 + C

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