A) \[\log \sec \left( \frac{\pi }{4}-x \right)+c\]
B) \[\log \cos \left( \frac{\pi }{4}+x \right)+c\]
C) \[\log \sin \left( \frac{\pi }{4}+x \right)+c\]
D) None of these
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{1-\tan x}{1+\tan x}\,dx}=\int_{{}}^{{}}{\tan \left( \frac{\pi }{4}-x \right)}\,dx\] \[=\int_{{}}^{{}}{\frac{\sin \left( \frac{\pi }{4}-x \right)}{\cos \left( \frac{\pi }{4}-x \right)}}\,dx=\log \cos \left( \frac{\pi }{4}-x \right)+c\] \[=\log \sin \left( \frac{\pi }{4}+x \right)+c\].
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