let-x-1-1-andf-n-x-sin-2narcsinx-1-prove-that-f-n-is-odd-and-calculate-f-n-0-and-f-n-1-2-solve-inside-0-1-f-n-x-0-3-prove-that-f-n-is-continue-derivable-on-1-1-and-calculate-f-n-

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Question Number 33981 by abdo imad last updated on 28/Apr/18

$$letx\in [-1,1]{andf}_n\left(x\right)=sin\left(2narcsinx\right)$$$$1)provethat{f}_{n}isoddandcalculate{f}_n\left(0\right)and{f}_n\left(1\right)$$$$2)solveinside\left[\overline{01}\right]f_n\left(x\right)=0$$$$3)provethat{f}_{n}iscontinue,derivableon[-1,1]and$$$$calculate{f}_n^{{}^{\prime }}\left(x\right)$$$$4)studythederivabilityof{f}_{n}at{1}^{-}and{(-1)}^{+}$$$$5)calculate{I}_n={\int }_0^1{f}_n\left(x\right)dx.$$

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