Definite-Integration Question 529
Question: The value of $ \int_a^{a+(\pi /2)}{({{\sin }^{4}}x+{{\cos }^{4}}x),dx} $ is
Options:
A) Independent of $ a $
B) $ a,{{( \frac{\pi }{2} )}^{2}} $
C) $ \frac{3\pi }{8} $
D) $ \frac{3\pi a^{2}}{8} $
Show Answer
Answer:
Correct Answer: C
Solution:
Since $ {{\sin }^{4}}x+{{\cos }^{4}}x $ is a periodic function with period $ \frac{\pi }{2}, $ therefore $ \int_a^{a+(\pi /2)}{({{\sin }^{4}}x+{{\cos }^{4}}x)dx} $ $ =\int_0^{\pi /2}{({{\sin }^{4}}x+{{\cos }^{4}}x)dx} $ $ =2\int_0^{\pi /2}{{{\sin }^{4}}x,dx=\frac{3\Gamma (5/2)\Gamma (1/2)}{2\Gamma ( \frac{4+0+2}{2} )}=\frac{3\pi }{8}} $ .
BETA
