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Question Number 36184 by prof Abdo imad last updated on 30/May/18

$$studytheconvergenceof{\int }_1^{+\mathrm{\infty }}\frac{cos\left(t\right)}{\sqrt{t}}dt$$

Commented by maxmathsup by imad last updated on 19/Aug/18

$$bypartsu=\frac{1}{\sqrt{t}}and{v}^{{}^{\prime }}=cost\Rightarrow$$$${\int }_1^{+\mathrm{\infty }}\frac{cost}{\sqrt{t}}dt={\left[\frac{sint}{\sqrt{t}}\right]}_1^{+\mathrm{\infty }}-{\int }_1^{+\mathrm{\infty }}(-\frac{1}{2t\sqrt{t}})sintdt$$$$=sin1+\frac{1}{2}{\int }_1^{+\mathrm{\infty }}\frac{sint}{t^{\frac{3}{2}}}dtbut$$$$\mid {\int }_1^{+\mathrm{\infty }}\frac{sint}{t^{\frac{3}{2}}}dt\mid ⩽{\int }_1^{+\mathrm{\infty }}\frac{dt}{t^{\frac{3}{2}}}andthisintevralconvergesbecause\frac{3}{2}>1.$$$$$$

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