What is the integral from 0 to 5 of (x-1)/(x^2-5x-6) ?

The integral from 0 to 5 of (x-1)/(x^2-5x-6) is -3/7 ln(6)

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integral from 0 to 5 of (x-1)/(x^2-5x-6)

\int_{0}^{5} \frac{x - 1}{x ^{2} - 5 x - 6} d x

- \frac{3}{7} ln (6)

Decimal

- 0.76789 \dots

Solution steps

\int_{0}^{5} \frac{x - 1}{x ^{2} - 5 x - 6} d x

Take the partial fraction of

\frac{x - 1}{x ^{2} - 5 x - 6} : \frac{2}{7 ( x + 1 )} + \frac{5}{7 ( x - 6 )}

= \int_{0}^{5} \frac{2}{7 ( x + 1 )} + \frac{5}{7 ( x - 6 )} 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}^{5} \frac{2}{7 ( x + 1 )} d x + \int_{0}^{5} \frac{5}{7 ( x - 6 )} d x

\int_{0}^{5} \frac{2}{7 ( x + 1 )} d x = \frac{2}{7} ln (6)

\int_{0}^{5} \frac{5}{7 ( x - 6 )} d x = - \frac{5}{7} ln (6)

= \frac{2}{7} ln (6) - \frac{5}{7} ln (6)

Simplify

\frac{2}{7} ln (6) - \frac{5}{7} ln (6) : - \frac{3}{7} ln (6)

= - \frac{3}{7} ln (6)

What is the integral from 0 to 5 of (x-1)/(x^2-5x-6) ?
The integral from 0 to 5 of (x-1)/(x^2-5x-6) is -3/7 ln(6)