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Question:

$\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x d x$

Solution:

Let $I=\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x d x$

$\int \operatorname{cosec} x d x=\log |\operatorname{cosec} x-\cot x|=\mathrm{F}(x)$

By second fundamental theorem of calculus, we obtain

$I=\mathrm{F}\left(\frac{\pi}{4}\right)-\mathrm{F}\left(\frac{\pi}{6}\right)$

$=\log \left|\operatorname{cosec} \frac{\pi}{4}-\cot \frac{\pi}{4}\right|-\log \left|\operatorname{cosec} \frac{\pi}{6}-\cot \frac{\pi}{6}\right|$

$=\log |\sqrt{2}-1|-\log |2-\sqrt{3}|$

$=\log \left(\frac{\sqrt{2}-1}{2-\sqrt{3}}\right)$