Question

${\int }_0^a$ $\ln (\cot a+\tan x)dx$ = , where $aϵ(0,\frac{\pi }{2})$ is

• A. $a\ln \left(\sin a\right)$
• B. $–a\ln \left(\sin a\right)$
• C. $–a\ln \left(\cos a\right)$
• D. none of these

Answer 1

The correct option is B $–a\ln \left(\sin a\right)$

$I={\int }_0^a\ln \frac{\cos (a-x)}{\sin a\cos x}dx$

$I={\int }_0^a\ln \frac{\cos x}{\sin a\cos (a-x)}dx$
Adding we get $2I={\int }_0^a\ln \frac{1}{(\sin a{)}^2}dx$
$={\int }_0^a-2\ln \left(\sin a\right)dx=-2a\ln \left(\sin a\right)\Rightarrow I=-a\ln \left(\sin a\right)$