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Question Number 169792 by mathocean1 last updated on 08/May/22

$$fiscontinuouson{\mathbb{R}}^{+}suchthat$$$${\int }_0^{+\mathrm{\infty }}f\left(t\right)dtisconvergent.$$$$Determinate\underset{x\to +\mathrm{\infty }}{lim}{\int }_x^{x^2}f\left(t\right)dt.$$

Answered by aleks041103 last updated on 08/May/22

$$letF\left(x\right)={\int }_0^{x}f\left(t\right)dt$$$$thenbyproblemconstruction\underset{x\to \mathrm{\infty }}{limF}\left(x\right)=a$$$$\underset{x\to \mathrm{\infty }}{lim}{\int }_x^{x^2}f\left(t\right)dt=\underset{x\to \mathrm{\infty }}{lim}[F\left(x^2\right)-F\left(x\right)]=$$$$=a-a=0$$$$\Rightarrow Ans.0$$

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