Integral Calculus Question 93
Question: If $ \int_{{}}^{{}}{(\sin 2x+\cos 2x)\ dx=\frac{1}{\sqrt{2}}\sin (2x-c)+a} $ , then the value of a and c is
[Roorkee 1978]
Options:
A) $ c=\pi /4 $ and $ a=k $ (an arbitrary constant)
B) $ c=-\pi /4 $ and $ a=\pi /2 $
C) $ c=\pi /2 $ and a is an arbitrary constant
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ \int_{{}}^{{}}{(\sin 2x+\cos 2x),dx}=-\frac{\cos 2x}{2}+\frac{\sin 2x}{2}+k $
$ =\frac{1}{\sqrt{2}}( \sin 2x\cos \frac{\pi }{4}-\cos 2x\sin \frac{\pi }{4} )+k $
$ =\frac{1}{\sqrt{2}}\sin ( 2x-\frac{\pi }{4} )+k $
$ \Rightarrow c=\frac{\pi }{4} $ and $ a=k, $ an arbitrary constant.
BETA
