Integral Calculus Question 293
Question: $ \int_{{}}^{{}}{\frac{dx}{\sin x+\sqrt{3}\cos x}}= $
Options:
A) $ \log \tan ( \frac{x}{2}+\frac{\pi }{2} )+c $
B) $ \frac{1}{2}\log \tan ( \frac{x}{2}+\frac{\pi }{6} )+c $
C) $ \log \cot ( \frac{x}{2}+\frac{\pi }{6} )+c $
D) $ \frac{1}{2}\log \cot ( \frac{x}{2}+\frac{\pi }{6} )+c $
Show Answer
Answer:
Correct Answer: B
Solution:
$ \int_{{}}^{{}}{\frac{dx}{\sin x+\sqrt{3}\cos x}}=\frac{1}{2}\int_{{}}^{{}}{\frac{dx}{\frac{\sin x}{2}+\frac{\sqrt{3}}{2}\cos x}} $ $ =\frac{1}{2}\int_{{}}^{{}}{\frac{dx}{\sin ( x+\frac{\pi }{3} )}}=\frac{1}{2}\int_{{}}^{{}}{cosec( x+\frac{\pi }{3} )} $ $ =\frac{1}{2}\log \tan ( \frac{x}{2}+\frac{\pi }{6} )+c. $
BETA
