Integral Calculus Question 281
Question: $ \int_{{}}^{{}}{\frac{dx}{\sin x+\cos x}}= $
[BIT Ranchi 1990; RPET 1997; Karnataka CET 1999; Orissa JEE 2004]
Options:
A) $ \log \tan ( \frac{\pi }{8}+\frac{x}{2} )+c $
B) $ \log \tan ( \frac{\pi }{8}-\frac{x}{2} )+c $
C) $ \frac{1}{\sqrt{2}}\log \tan ( \frac{\pi }{8}+\frac{x}{2} )+c $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
$ \int_{{}}^{{}}{\frac{dx}{\sin x+\cos x}}=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\frac{dx}{\sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4}}} $ $ =\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\text{cosec }( x+\frac{\pi }{4} ),dx=\frac{1}{\sqrt{2}}\log \tan ( \frac{\pi }{8}+\frac{x}{2} )}+c. $
BETA
