Definite Integration Question 467
Question: For any integer $ n, $ the integral $ \int_0^{\pi }{{e^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)xdx=} $
[MNR 1982]
Options:
A) $ -1 $
B) 0
C) 1
D) $ \pi $
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ f(x)=\int_0^{\pi }{{e^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x.dx} $
Since $ \cos (2n+1)(\pi -x)=\cos [(2n+1)\pi -(2n+1)x] $
$ =-\cos (2n+1)x $ and $ {{\sin }^{2}}(\pi -x)={{\sin }^{2}}x $
Hence by the property of definite integral, $ \int_0^{\pi }{{e^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)xdx=0} $ , $ [f(2a-x)=-f(x)] $ .
BETA
