Question

निम्नलिखित प्रश्न में निश्चित समाकलन का मान ज्ञात कीजिए।

${\displaystyle {\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}} \text{cosec} x dx}$

Answer 1

${\displaystyle {\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}} \cos ecx dx}$

${\displaystyle ={\left[\log (\text{cosec} x+\cot x)\right]}_{\frac{\pi }{6}}^{\frac{\pi }{4}}}$

${\displaystyle =\log (\text{cosec} \frac{\pi }{4}-\text{cot} \frac{\pi }{4})-\log (\text{cosec} \frac{\pi }{6}-\text{cot}\frac{\pi }{6})}$

${\displaystyle =\log (\sqrt{2}-1)-\log (2-\sqrt{3})}$

${\displaystyle =\log \frac{\sqrt{2}-1}{2-\sqrt{3}}}$

${\displaystyle \because \log \frac{m}{n}}$

= log m = log n