Integral Calculus Question 91
Question: $ \int \frac{dx}{\cos x+\sqrt{3}\sin x} $ is equal to
Options:
A) $ \frac{1}{2}\log ,\tan ( \frac{x}{2}+\frac{\pi }{12} )+c $
B) $ \frac{1}{2}\log \tan ( \frac{x}{2}-\frac{\pi }{12} )+c $
C) $ \log \tan ( \frac{x}{2}+\frac{\pi }{12} )+c $
D) $ \log ,\tan ( \frac{x}{2}-\frac{\pi }{12} )+c $
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Answer:
Correct Answer: A
Solution:
[a] $ \int{\frac{dx}{\cos +\sqrt{3}\sin x}=\frac{1}{2}\int{\frac{dx}{\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x}}} $ $ =\frac{1}{2}\int{\frac{dx}{\cos ( x-\frac{\pi }{3} )}}=\frac{1}{2}\int{\sec ( x-\frac{\pi }{3} )dx} $ $ =\frac{1}{2}\log \tan ( \frac{x}{2}-\frac{\pi }{6}+\frac{\pi }{4} )+c $ $ =\frac{1}{2}\log \tan ( \frac{x}{2}+\frac{\pi }{12} )+c $
BETA
