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

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

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

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

ln ∣ x - 1 ∣ + arctan (x) + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= ln ∣ x - 1 ∣ + arctan (x)

Add a constant to the solution

= ln ∣ x - 1 ∣ + arctan (x) + C

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