Definite Integration Question 462
Question: The value of the integral $ \int _{-\pi /4}^{\pi /4}{{{\sin }^{-4}}x}dx $ is
[IIT Screening; MP PET 2003]
Options:
A) 3/2
B) -8/3
C) 3/8
D) 8/3
Show Answer
Answer:
Correct Answer: B
Solution:
$ \int _{-\pi /4}^{\pi /4}{{{\sin }^{-4}}xdx=2\int_0^{\pi /4}{\frac{{{\cos }^{4}}x}{{{\sin }^{4}}x}{{\sec }^{4}}xdx}}=2\int_0^{\pi /4}{\frac{{{\sec }^{4}}xdx}{{{\tan }^{4}}x}} $
Put $ \tan x=t $ , we get $ 2\int_0^{1}{\frac{1+t^{2}}{t^{4}}dt} $
$ =2[ \int_0^{1}{{t^{-4}}dt+\int_0^{1}{{t^{-2}}dt}} ] $
$ =2[ | -\frac{1}{3t^{3}} |_0^{1}+| -\frac{1}{t} |_0^{1} ]=-\frac{8}{3} $ .
BETA
