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Question Number 33981 by abdo imad last updated on 28/Apr/18
![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 ^′ (x) 4) study the derivability of f_n at 1^− and (−1)^+ 5)calculate I_n = ∫_0 ^1 f_n (x)dx .](Q33981.png)
$${let}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right]\:{andf}_{{n}} \left({x}\right)={sin}\left(\mathrm{2}{narcsinx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}_{{n}} {is}\:{odd}\:{and}\:{calculate}\:{f}_{{n}} \left(\mathrm{0}\right)\:{and}\:{f}_{{n}} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){solve}\:{inside}\:\left[\bar {\mathrm{0}1}\right]\:\:{f}_{{n}} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{f}_{{n}} \:{is}\:{continue},{derivable}\:{on}\left[−\mathrm{1},\mathrm{1}\right]\:{and} \\ $$$${calculate}\:{f}_{{n}} ^{'} \left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{study}\:{the}\:{derivability}\:{of}\:{f}_{{n}} \:{at}\:\mathrm{1}^{−} \:\:{and}\:\left(−\mathrm{1}\right)^{+} \\ $$$$\left.\mathrm{5}\right){calculate}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right){dx}\:. \\ $$