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

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

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

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

- \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3} ln x^{2} - x + 1 + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{2}{3 ( x + 1 )} d x = \frac{2}{3} ln ∣ x + 1 ∣

\int \frac{2 x - 1}{3 ( x ^{2} - x + 1 )} d x = \frac{1}{3} ln x^{2} - x + 1

= - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3} ln x^{2} - x + 1

Add a constant to the solution

= - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3} ln x^{2} - x + 1 + C

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