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

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

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

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

ln ∣ x - 2 ∣ + 2 arctan (x) + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= ln ∣ x - 2 ∣ + 2 arctan (x)

Add a constant to the solution

= ln ∣ x - 2 ∣ + 2 arctan (x) + C

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