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

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

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

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

- \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{3 ( x - 1 )} d x = \frac{1}{3} ln ∣ x - 1 ∣

\int \frac{1}{3 ( x + 2 )} d x = \frac{1}{3} ln ∣ x + 2 ∣

= - \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + C

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