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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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$ I= \int \limits_0^{\frac{\pi}{2}} \cos ^3 x dx$

\[\begin {array} {1 1} (a)\;\frac{1}{3} \\ (b)\;1 \\ (c)\;\frac{2}{3} \\ (d)\;0 \end {array}\]
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1 Answer

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$I= \int \limits_0^{\frac{\pi}{2}} \cos ^3 x dx$
$\quad= \int \limits_0^{\frac{\pi}{2}} \cos x (\cos ^2 x) dx$
$\quad= \int \limits_0^{\frac{\pi}{2}} \cos x (1-\sin ^2 x) dx$
$\quad= \int \limits_0^{\frac{\pi}{2}} \cos x dx -\int \limits_0^{\frac{\pi}{2}} \cos x \sin ^2 x dx$
$\sin x =t \quad \cos x dx=dt$
$\qquad= \sin x \bigg]_0^{\pi/2} - \int \limits _0^1 t^2 dt$
$\qquad=1-\large\frac{1}{3} $
$\qquad=\large\frac{2}{3}$
Hence c is the correct answer.
answered Dec 21, 2013 by meena.p
 
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