Definite Integration Question 190
Question: $ \int_0^{\pi /4}{{}}\sec x\log (\sec x+\tan x)dx= $
Options:
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}} $
Show Answer
Answer:
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 xdx=dt $
$ \Rightarrow I=\int_0^{\log (\sqrt{2}+1)}{tdt=[ \frac{t^{2}}{2} ]}_0^{\log (\sqrt{2}+1)}=\frac{{{[\log (\sqrt{2}+1)]}^{2}}}{2} $ .
BETA
