A) \[\frac{1}{2}{{[\log (1+\sqrt{2})]}^{2}}\]
B) \[{{[\log (1+\sqrt{2})]}^{2}}\]
C) \[\frac{1}{2}{{[\log (\sqrt{2}-1)]}^{2}}\]
D) \[\frac{1}{2}{{[\log (\sqrt{2}-1)]}^{2}}\]
Correct Answer: A
Solution :
\[I=\int_{0}^{\pi /4}{\sec x\log (\sec x+\tan x)dx}\] Put \[\log (\sec x+\tan x)=t\Rightarrow \sec x\,dx=dt\] \[\Rightarrow I=\int_{0}^{\log (\sqrt{2}+1)}{t\,dt=\left[ \frac{{{t}^{2}}}{2} \right]}_{0}^{\log (\sqrt{2}+1)}=\frac{{{[\log (\sqrt{2}+1)]}^{2}}}{2}\].
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