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Home  >>  CBSE XII  >>  Math  >>  Integrals
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By using the properties of definite integrals,evaluate the integral\[\int\limits_0^{2\pi}\cos^5x\;dx\]

$\begin{array}{1 1} \frac{\pi}{2} \\ \frac{\pi}{4} \\ \pi \\ 0 \end{array} $

Can you answer this question?
 
 

1 Answer

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Toolbox:
  • (i)$\int\limits_a^b f(x)dx=F(b)-F(a)$
  • $\int \limits_0^{2a}f(x)dx=2\int f(x)dx$if $(2a-x)=f(x)$
  • $=0\;if\;(2a-x)=-f(x)$
Given $I=\int\limits_0^{2\pi}\cos^5x\;dx$
 
Let us consider $\int\limits_0^{2\pi}\cos^5x$
 
$2\int \limits_0^\pi \cos ^5(\pi-x)=-cos^5(\pi-x).$
 
Since $\int \limits_0^a f (2a-x)=f(x)=0$
 
$2\int \limits_0^\pi \cos^5(\pi-x)=0$

 

answered Feb 14, 2013 by meena.p
 

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