A) \[\frac{5}{3}\]
B) \[\frac{5}{4}\]
C) \[\frac{1}{3}\]
D) \[\frac{1}{6}\]
Correct Answer: D
Solution :
We have, \[I=\int_{0}^{\pi /8}{{{\cos }^{3}}4\theta \,d\theta }\] \[=\int_{0}^{\pi /8}{{{\cos }^{2}}4\theta \,\cos 4\theta }\,d\theta \] \[=\int_{0}^{\pi /8}{\left( \frac{1+\cos 8\,\theta }{2} \right)\,\cos 4\theta }\,d\theta \] \[=\frac{1}{2}\int_{0}^{\pi /8}{\,\cos 4\theta }\,d\theta +\frac{1}{2}\int_{0}^{\pi /8}{\,\cos 8\theta \,\cos 4\,\theta \,d\,\theta }\] \[=\frac{1}{2}\left[ \sin \frac{4\theta }{2} \right]_{0}^{\pi /8}+{{I}_{1}}\] ?. (i) Now, \[{{I}_{1}}=\frac{1}{2}\int_{0}^{\pi /8}{\,\cos 8\theta }\cos 4\,\theta \,d\,\theta \] \[=\frac{1}{2}\left[ \cos 8\theta \frac{\sin \,4\theta }{4} \right]_{0}^{\pi /8}\] \[-\frac{1}{2}\int_{0}^{\pi /8}{(-8\,\cos \,8\theta )\frac{\sin \,4\theta }{4}d\theta }\] \[=-\frac{1}{8}+\int_{0}^{\pi /8}{\sin \,8\theta \,\,\sin \,4\theta \,d\theta }\] \[=-\frac{1}{8}+\left[ \sin 8\theta \left( -\frac{\cos 4\theta }{4} \right) \right]_{0}^{\pi /8}\] \[-\int_{0}^{\pi /8}{8\,\cos \,8\theta \left( -\frac{\cos 4\theta }{4} \right)d\theta }\] \[=-\frac{1}{2}+0+2\,\int_{0}^{\pi /8}{\cos 8\theta \cos 4\theta \,d\theta }\] \[{{I}_{1}}=-\frac{1}{8}+4{{I}_{1}}\] \[\Rightarrow \] \[=3\,{{I}_{1}}=-\frac{1}{8}\] \[{{I}_{1}}=1/24\] ... (ii) from equation (i) and (ii), we get \[I=\frac{1}{8}+\frac{1}{24}\] \[=\frac{3+1}{24}=\frac{4}{24}=\frac{1}{6}\]
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