Evaluate the following definite integrals: _ 6 ^ 4 cosecx dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{4}}\text{cosec}\text{x}\text{ dx}$

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Answer

Let $\text{I}=\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{4}}\text{cosec}\text{x}\text{ dx}$
$\int\text{cosec}\text{x dx}=\log|\text{cosecx}-\cot\text{x}|=\text{F}(\text{x})$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}\Big(\frac{\pi}{4}\Big)-\text{F}\Big(\frac{\pi}{6}\Big)$
$=\log\Big|\text{cosec}\frac{\pi}{4}-\cot\frac{\pi}{4}\Big|-\log\Big|\text{cosec}\frac{\pi}{6}-\cot\frac{\pi}{6}\Big|$
$=\log\big|\sqrt{2}-1\big|-\log\big|2-\sqrt{3}\big|$
$=\log\bigg(\frac{\sqrt{2}-1}{2-\sqrt{3}}\bigg)$

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