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

l e t x \in [- 1, 1] {a n d f}_{n} (x) = s i n (2 n a r c s i n x)

1) p r o v e t h a t f_{n} i s o d d a n d c a l c u l a t e f_{n} (0) a n d f_{n} (1)

2) s o l v e i n s i d e [\bar{01}] f_{n} (x) = 0

3) p r o v e t h a t f_{n} i s c o n t i n u e, d e r i v a b l e o n [- 1, 1] a n d

c a l c u l a t e f_{n}^{^{'}} (x)

4) s t u d y t h e d e r i v a b i l i t y o f f_{n} a t 1^{-} a n d {(- 1)}^{+}

5) c a l c u l a t e I_{n} = \int_{0}^{1} f_{n} (x) d x .