A) Independent of \[a\]
B) \[a\,{{\left( \frac{\pi }{2} \right)}^{2}}\]
C) \[\frac{3\pi }{8}\]
D) \[\frac{3\pi {{a}^{2}}}{8}\]
Correct Answer: C
Solution :
Since \[{{\sin }^{4}}x+{{\cos }^{4}}x\]is a periodic function with period \[\frac{\pi }{2},\] therefore \[\int_{a}^{a+(\pi /2)}{({{\sin }^{4}}x+{{\cos }^{4}}x)\text{ }dx}\] \[=\int_{0}^{\pi /2}{({{\sin }^{4}}x+{{\cos }^{4}}x)dx}\] \[=2\int_{0}^{\pi /2}{{{\sin }^{4}}x\,dx=\frac{3\Gamma (5/2)\Gamma (1/2)}{2\Gamma \left( \frac{4+0+2}{2} \right)}=\frac{3\pi }{8}}\].
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