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The value of $\int\limits_{0}^{\pi/4}\cos^{3}2x dx$ is

\[\begin{array}{1 1}(1)\frac{2}{3}&(2)\frac{1}{3}\\(3)0&(4)\frac{2\pi}{3}\end{array}\]

Can you answer this question?
 
 

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$\int\limits_{0}^{\pi/4}\cos^{3}2x dx$
Put $2x=\theta$
$2x=d \theta$
$dx =\large\frac{1}{2}$$ d \theta$
When $x=0=> \theta=0$
When $x=\large\frac{\pi}{2}$$=> \theta=2 \times \large\frac{\pi}{4}=\frac{\pi}{2}$
$\int\limits_{0}^{\pi/4}\cos^{3}2x dx= \int\limits_{0}^{\pi/2}\cos^{3} \theta \large\frac{1}{2} d \theta$
$\qquad= \large\frac{1}{2} \int\limits_{0}^{\pi/2}$$ \cos^{3} \theta d \theta$
$\qquad= \large\frac{1}{2} . \frac{3-1}{3} .1$
$\qquad= \large\frac{1}{2} . \frac{2}{3} = \frac{1}{3}$
Hence 2 is the correct answer.
answered May 21, 2014 by meena.p
 

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