A) \[\frac{\pi }{4}\]
B) \[\frac{\pi }{2}\]
C) \[\frac{3\pi }{2}\]
D) \[\frac{\pi }{4}\]
E) \[\frac{2\pi }{3}\]
Correct Answer: A
Solution :
Let\[I=\int_{-\pi }^{\pi }{\frac{{{\sin }^{4}}xdx}{{{\sin }^{4}}x+{{\cos }^{4}}x}}\] \[=4\int_{0}^{\pi }{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx\] \[I=4\int_{0}^{\pi /2}{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx\] ?.(i) \[I=4\int_{0}^{\pi /2}{\frac{{{\cos }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx\] ?..(ii) On adding Eqs. (i) and (ii), we get \[2I=4\int_{0}^{\pi /2}{1.}\,dx=2\pi \]
You need to login to perform this action.
You will be redirected in
3 sec
