How to check convergence of $$\int_{-\infty}^{2}\frac{e^x}{\sqrt[3]{x^2-1}}\ dx \ ?$$ No convergent integral that could bound this one came to my mind, nor any which would fit asymptotic criterium. Any hint maybe?
2026-04-02 21:45:19.1775166319
Tricky improper integral
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As Daniel Fischer said there are 3 critical points: $-\infty$ and $\pm1$. Note $\pm1$ have same character. So we should observe 2 cases: $$ \int_{-\infty}^{-2}\frac{e^x}{\sqrt[3]{x^2-1}}\ dx < C_1 \int_{-\infty}^{-2}e^x\ dx $$ and $$ \int_{-2}^{2}\frac{e^x}{\sqrt[3]{x^2-1}}\ dx < C_2 \int_{-2}^{-2}\frac{1}{\sqrt[3]{x^2-1}}\ dx, $$ where $C_1$ and $C_2$ some constants. We have proved convergence.