Question

The value of $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{4}}{\int }}\text{cosec}xdx$ is;

• A. $\ln \left|\frac{\sqrt{2}-1}{2-\sqrt{3}}\right|$
• B. $\ln |\sqrt{2}-1|$
• C. $\ln |2-\sqrt{3}|$
• D. $\ln \left|\frac{1}{\sqrt{2}-1}\right|$

Answer 1

The correct option is A $\ln \left|\frac{\sqrt{2}-1}{2-\sqrt{3}}\right|$

Let $I=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{4}}{\int }}\text{cosec}xdx$
$I={\left[\ln |\text{cosec}x-\cot x|\right]}_{\frac{\pi }{6}}^{\frac{\pi }{4}}[\because \int \text{cosec}xdx=\ln |\text{cosec}x-\cot x|+c]=\ln |\text{cosec}\frac{\pi }{4}-\cot \frac{\pi }{4}|-\ln |\text{cosec}\frac{\pi }{6}-\cot \frac{\pi }{6}|=\ln |\sqrt{2}-1|-\ln |2-\sqrt{3}|=\ln \left|\frac{\sqrt{2}-1}{2-\sqrt{3}}\right|$