Question

${\int }_0^{\frac{\pi }{4}}\log \left(\frac{\sin x+\cos x}{\cos x}\right)dx$

• A. $\frac{\pi }{4}\log 2$
• B. $\frac{\pi }{2}\log 2$
• C. $\frac{\pi }{8}\log 2$
• D. $\log 2$

Answer 1

The correct option is C $\frac{\pi }{8}\log 2$

$I={\int }_0^{\frac{\pi }{4}}\log \left(\frac{\sin x+\cos x}{\cos x}\right)dx={\int }_0^{\frac{\pi }{4}}\log (1+\tan x)dx={\int }_0^{\frac{\pi }{4}}\log (1+\tan (\frac{\pi }{4}-x\left)\right)dx={\int }_0^{\frac{\pi }{4}}(\log 2-\log (1+\tan x\left)\right)dx\Rightarrow 2I={\int }_0^{\frac{\pi }{4}}\log 2dxI=\frac{\pi }{8}\log 2$