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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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