Question:
$\int_{2}^{3} \frac{d x}{x^{2}-1}$
Solution:
Let $I=\int_{2}^{3} \frac{d x}{x^{2}-1}$
$\int \frac{d x}{x^{2}-1}=\frac{1}{2} \log \left|\frac{x-1}{x+1}\right|=\mathrm{F}(x)$
By second fundamental theorem of calculus, we obtain
$I=\mathrm{F}(3)-\mathrm{F}(2)$
$=\frac{1}{2}\left[\log \left|\frac{3-1}{3+1}\right|-\log \left|\frac{2-1}{2+1}\right|\right]$
$=\frac{1}{2}\left[\log \left|\frac{2}{4}\right|-\log \left|\frac{1}{3}\right|\right]$
$=\frac{1}{2}\left[\log \frac{1}{2}-\log \frac{1}{3}\right]$
$=\frac{1}{2}\left[\log \frac{3}{2}\right]$
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