Question

Assertion :${\int }_0^{\pi /2}x\cot xdx=\frac{\pi }{2}\log 2$ Reason: ${\int }_0^{\pi /2}\log \sin xdx=-\frac{\pi }{2}\log 2$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

As ${\int }_0^{\frac{\pi }{2}}\log \sin x=-\frac{\pi }{2}\log 2$
${\int }_0^{\frac{\pi }{2}}x\cot xdx={\left[x\log \sin x\right]}_0^{\frac{\pi }{2}}-{\int }_0^{\frac{\pi }{2}}\log \sin xdx$
$=0-[-\frac{\pi }{2}\log 2]=\frac{\pi }{2}\log 2$