---
alias: [ "critère de Bertrand" ]
---
up:: [[intégration généralisée]] 
title:: "$\displaystyle\int_{0}^{m<e} \frac{1}{t^{\alpha}|\ln(t)|^{\beta}} \, dx$ et $\displaystyle\int_{m>e}^{+\infty} \frac{1}{t^{\alpha}(\ln(t))^{\beta}} \, dt$"
description:: "$\displaystyle\int_{0}^{m<e} \frac{1}{t^{\alpha}|\ln(t)|^{\beta}} \, dx \text{ CV} \iff\begin{cases} \alpha<1\\ \text{ou}\\ \alpha=1 \text{ et } \beta>1 \end{cases}$", "$\displaystyle\int_{m>e}^{+\infty} \frac{1}{t^{\alpha}(\ln(t))^{\beta}} \, dt \text{ CV} \iff\begin{cases} \alpha > 1\\ \text{ou}\\ \alpha = 1 \text{ et } \beta > 1 \end{cases}$"
#maths/analyse 

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![[intégrales de Bertrand 2022-11-23 16.18.37.excalidraw|100%]]


$\displaystyle\int_{0}^{m<e} \frac{1}{t^{\alpha}|\ln(t)|^{\beta}} \, dx$ et $\displaystyle\int_{m>e}^{+\infty} \frac{1}{t^{\alpha}(\ln(t))^{\beta}} \, dt$