Question

Evaluate : $\int\limits^3_2 {\frac{x}{x^{2}-1}} \, dx$

Answer 1

Solution:

${x}^{2} - 1 = t \\ \\ 2x \: dx = dt \\ \\ x \: dx = \frac{dt}{2}$
now put this value in the integration

$\int\limits^3_2 {\frac{x}{x^{2}-1}} \, dx = \int\limits^3_2 \: \frac{1}{2} \times \frac{1}{t} \: dt \\ \\ = \frac{1}{2} \int\limits^3_2 \: \frac{1}{t} \: dt \\ \\ = [\frac{1}{2} \: log \: t] \: \: put \: limits \: now \\ \\ = \frac{1}{2} log(3) - \frac{1}{2} log(2) \\ \\ = \frac{1}{2} log( \frac{3}{2} )$
So

$\int\limits^3_2 {\frac{x}{x^{2}-1}} \, dx \: = \frac{1}{2} log( \frac{3}{2} )$