Question Number 33339 by prof Abdo imad last updated on 14/Apr/18
$${find}\:\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {xdx}\: \\ $$
Commented by prof Abdo imad last updated on 25/Apr/18
$${let}\:{put}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {x}\:{dx} \\ $$$$=\:\int_{{R}} \:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {x}\:\:\chi_{\left[\bar {\mathrm{0}},{n}\left[\right.\right.} \left({x}\right){dx}\:\:{but} \\ $$$${f}_{{n}} \left({x}\right)=\:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {x}\:\chi_{\left[\mathrm{0},{n}\left[\right.\right.} \left({x}\right)\:\rightarrow^{{c}.{s}} \:\:\mathrm{0}\:\:{so} \\ $$$${lim}_{{n}\rightarrow+\infty} \:{I}_{{n}} \:\:=\mathrm{0} \\ $$