What is the integral from 0 to 1 of (x+2)/(x^2-x-2) ?

The integral from 0 to 1 of (x+2)/(x^2-x-2) is -5/3 ln(2)

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integral from 0 to 1 of (x+2)/(x^2-x-2)

\int_{0}^{1} \frac{x + 2}{x ^{2} - x - 2} d x

- \frac{5}{3} ln (2)

Decimal Notation

- 1.15524 \dots

Solution steps

\int_{0}^{1} \frac{x + 2}{x ^{2} - x - 2} d x

Take the partial fraction of

\frac{x + 2}{x ^{2} - x - 2} : - \frac{1}{3 ( x + 1 )} + \frac{4}{3 ( x - 2 )}

= \int_{0}^{1} - \frac{1}{3 ( x + 1 )} + \frac{4}{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_{0}^{1} \frac{1}{3 ( x + 1 )} d x + \int_{0}^{1} \frac{4}{3 ( x - 2 )} d x

\int_{0}^{1} \frac{1}{3 ( x + 1 )} d x = \frac{1}{3} ln (2)

\int_{0}^{1} \frac{4}{3 ( x - 2 )} d x = - \frac{4}{3} ln (2)

= - \frac{1}{3} ln (2) - \frac{4}{3} ln (2)

Simplify

- \frac{1}{3} ln (2) - \frac{4}{3} ln (2) : - \frac{5}{3} ln (2)

= - \frac{5}{3} ln (2)

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What is the integral from 0 to 1 of (x+2)/(x^2-x-2) ?
The integral from 0 to 1 of (x+2)/(x^2-x-2) is -5/3 ln(2)