Definite Integration Question 338
Question: $ \int_0^{1}{\frac{x^{7}}{\sqrt{1-x^{4}}}dx} $ is equal to
[AMU 2000]
Options:
A) 1
B) $ \frac{1}{3} $
C) $ \frac{2}{3} $
D) $ \frac{\pi }{3} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ I=\int_0^{1}{\frac{x^{7}}{\sqrt{1-x^{4}}}dx=\int_0^{1}{\frac{x^{6}xdx}{\sqrt{1-x^{4}}}}} $
Put $ x^{2}=\sin \theta $
$ \Rightarrow 2xdx=\cos \theta d\theta $
$ I=\frac{1}{2}\int_0^{\pi /2}{\frac{{{\sin }^{3}}\theta .\cos \theta d\theta }{\cos \theta }}=\frac{1}{2}\int_0^{\pi /2}{{{\sin }^{3}}\theta d\theta } $
$ =\frac{1}{2}\frac{\Gamma 2\Gamma (1/2)}{2.\Gamma (5/2)}=\frac{\Gamma ( \frac{1}{2} )}{4.\frac{3}{2}.\frac{1}{2}.\Gamma ( \frac{1}{2} )}=\frac{1}{3} $ .
BETA
