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

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

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

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

\frac{x ^{2}}{2} - x + ln x^{2} + x - 2 + C

Solution steps

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

Long division

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

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

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

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

= \frac{x ^{2}}{2} - x + ln x^{2} + x - 2

Add a constant to the solution

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

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