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Evaluate:$\int \limits_{-\pi}^{\pi} x^{20} \sin ^9x dx $

$\begin{array}{1 1}(A)\;-1\\(B)\;\large\frac{-1}{2}\\(C)\;\large\frac{\pi}{2}\\(D)\;0\end{array}$

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  • $\int \limits _a^b f(x)dx=F(b)-F(a)$
  • $\int \limits _{-a}^a f(x)dx=0$ if f(x) is an odd function
  • If $f(-x)=-f(x),$ then the function is an odd function
Given $I=\int \limits_{-\pi}^{\pi} x^{20} \sin ^9x dx $
Let $ x^{20} \sin ^9x =f(x)$
if x is repaced by -x,
then $f(-x)=(-x)^{20}\; \sin^9(-x)$
But $ \sin (-x)=-\sin x$
$=x^{20} \times -\sin ^9(x)$
$=-[x^{20}.\sin ^9 x]$
Hence $f(-x)=-f(x)$
This is an odd function
we know if the given function is an odd funtion, then
$\int \limits _{-a}^a f(x)dx=0$
Hence I=0
answered Apr 1, 2013 by meena.p
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