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Question Number 133091 by metamorfose last updated on 18/Feb/21

$$\underset{x\to +\mathrm{\infty }}{\lim }{\int }_0^1{(1-t^2)}^{3}cos\left(xt\right)dt...?$$

Answered by mnjuly1970 last updated on 18/Feb/21

$$answer:=0$$$$reiman-lebesguetheorem$$$$fiscontinuoueson[0,1]$$$${lim}_{x\to \mathrm{\infty }}{\int }_0^{1}f\left(t\right)sin\left(xt\right)dt=0$$$$$$

Answered by mathmax by abdo last updated on 20/Feb/21

$$\mathrm{let}\mathrm{f}\left(\mathrm{t}\right)\mathrm{is}\mathrm{pritive}\mathrm{of}{(1-{\mathrm{t}}^2)}^3\mathrm{by}\mathrm{parts}\mathrm{we}\mathrm{get}$$$${\int }_0^1{(1-{\mathrm{t}}^2)}^3\cos \left(\mathrm{xt}\right)\mathrm{dt}={\left[\frac{\mathrm{f}\left(\mathrm{t}\right)}{\mathrm{x}}\sin \left(\mathrm{xt}\right)\right]}_0^1-{\int }_0^1\frac{\mathrm{f}\left(\mathrm{t}\right)}{\mathrm{x}}\sin \left(\mathrm{xt}\right)\mathrm{dt}$$$$=\frac{\mathrm{f}\left(1\right)\sin \left(\mathrm{x}\right)}{\mathrm{x}}-\frac{1}{\mathrm{x}}{\int }_0^1\mathrm{f}\left(\mathrm{t}\right)\sin \left(\mathrm{xt}\right)\mathrm{dt}\Rightarrow \mathrm{for}\mathrm{x}>0\mid {\int }_0^1(...)\mathrm{dt}\mid$$$$⩽\frac{\mid \mathrm{f}\left(1\right)\mid }{\mathrm{x}}+\frac{1}{\mathrm{x}}{\int }_0^1\mid \mathrm{f}\left(\mathrm{t}\right)\mid \mathrm{dt}\to 0(\mathrm{x}\to +\mathrm{\infty })\Rightarrow$$$${\lim }_{\mathrm{x}\to +\mathrm{\infty }}{\int }_0^1{(1-{\mathrm{t}}^2)}^3\cos \left(\mathrm{xt}\right)\mathrm{dt}=0$$

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