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Question Number 121314 by Bird last updated on 06/Nov/20

$$find{lim}_{n\to +\mathrm{\infty }}{\int }_0^n\frac{cos\left(nx\right)}{x^2+n^2}dx$$

Answered by mathmax by abdo last updated on 06/Nov/20

$${\mathrm{I}}_{\mathrm{n}}={\int }_0^{\mathrm{n}}\frac{\cos \left(\mathrm{nx}\right)}{{\mathrm{x}}^2+{\mathrm{n}}^2}\mathrm{dx}\mathrm{changement}\mathrm{x}=\mathrm{nt}\mathrm{give}$$$${\mathrm{I}}_{\mathrm{n}}={\int }_0^1\frac{\cos \left({\mathrm{n}}^2\mathrm{t}\right)}{{\mathrm{n}}^2(1+{\mathrm{t}}^2)}\mathrm{ndt}=\frac{1}{\mathrm{n}}{\int }_0^1\frac{\cos \left({\mathrm{n}}^2\mathrm{t}\right)}{{\mathrm{t}}^2+1}\mathrm{dt}\Rightarrow \mid {\mathrm{I}}_{\mathrm{n}}\mid ⩽\frac{1}{\mathrm{n}}{\int }_0^1\frac{\mathrm{dt}}{{\mathrm{t}}^2+1}\Rightarrow$$$$\mid {\mathrm{I}}_{\mathrm{n}}\mid ⩽\frac{\pi }{4\mathrm{n}}\to 0\Rightarrow {\lim }_{\mathrm{n}\to +\mathrm{\infty }}{\mathrm{I}}_{\mathrm{n}}=0$$

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