Question

${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}cotxdx=$

• A. $lo{g}^2$
• B. $\sqrt{log}2$
• C. log $\sqrt{2}$
• D. $\frac{1}{log2}$

Answer 1

The correct option is C log $\sqrt{2}$

${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\cot xdx$

As we know that,

$\int \cot xdx=\log \sin x+C$

Therefore,

${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\cot xdx$

$={\left[\log \sin x\right]}_{\frac{\pi }{4}}^{\frac{\pi }{2}}$

$=\log \left(\sin \frac{\pi }{2}\right)-\ln \left(\sin \frac{\pi }{4}\right)$

$=\log 1-\log \frac{1}{\sqrt{2}}$

$=0-(\log 1-\log \sqrt{2})(\because \log 1=0)$

$=\log \sqrt{2}$

Hence the correct answer is $\left(C\right)\log \sqrt{2}$.