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

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

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

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

- ln ∣ x + 1 ∣ + ln x^{2} + 1 + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= - ln ∣ x + 1 ∣ + ln x^{2} + 1

Add a constant to the solution

= - ln ∣ x + 1 ∣ + ln x^{2} + 1 + C

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