Integral Calculus Question 175
Question: What is $ \int\limits_{-\frac{\pi }{6}}^{\frac{\pi }{6}}{\frac{{{\sin }^{5}}x{{\cos }^{3}}x}{x^{4}}}dx $ equal to?
Options:
A) $ \frac{\pi }{2} $
B) $ \frac{\pi }{4} $
C) $ \frac{\pi }{8} $
D) 0
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Let $ f(x)=\frac{{{\sin }^{5}}x{{\cos }^{3}}x}{x^{4}} $ $ f(-x)=\frac{{{\sin }^{5}}(-x){{\cos }^{3}}(-x)}{{{(-x)}^{4}}} $ $ =\frac{-{{\sin }^{5}}x{{\cos }^{3}}x}{x^{4}}=-f(x) $
$ \Rightarrow f(x) $ is an odd function. Hence, $ \int\limits_{-\frac{\pi }{6}}^{\frac{\pi }{6}}{\frac{{{\sin }^{5}}x{{\cos }^{3}}x}{x^{4}}dx=0} $
BETA
