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Evaluate:\[\int_{-\pi}^\pi (\sin ^{-1}x+x^{295})\]

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  • (i) $\int \limits_a^b f(x)dx=F(b)-F(a)$
  • (ii) If $f(-x) =-f(x),$ then the function is an odd function. hence for an odd function $\int \limits_{-a}^af(x)dx=0$
Given $I=\int_{-\pi}^\pi (\sin ^{-1}x+x^{295})_{dx}^{-a}$
Consider $ \sin ^{-1}(x)+x^{295}$ Replace x by -x
$ \sin ^{-1}(-x)+(-x)^{295}$
This is equal to 0,hence it is an odd function
We know $\int \limits_{-a}^af(x)dx=0$ if f(x) is an odd function
Therefore $I=\int_{-\pi}^\pi (\sin ^{-1}(x)+x^{295}){dx}=0$


answered Mar 14, 2013 by meena.p
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