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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int 3 d x = 3 x

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

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

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

Add a constant to the solution

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

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