# Questions tagged [improper-integrals]

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or $\infty$ or $-\infty$, or as both endpoints approach limits.

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### Show $\int_0^\infty f(x) \cos(ax) dx \ge 0$ if $f$ is convex and $a \in \mathbb R$.

Let $f$ be a convex function and $a \in \mathbb R$. Is it true that $$\int_0^\infty f(x) \cos(ax) dx \ge 0$$ holds? Of course, this would be true if we replaced $\cos$ by $|\cos|$, but I guess that ...
### Improper integral $\int_0^{+\infty} \frac{\arctan x}{\sqrt{x}} \sin(2x) dx$
Check absolute and conditional convergence of following improper integral $$\int_0^{+\infty} \frac{\arctan x}{\sqrt{x}} \sin(2x) dx$$ Can someone please check if my solution is alright. Here is what I ...
I am trying to find the solution of the integral: $$\int_a^{\infty}\left(\frac{d}{dt}Ai(t/3^{1/3})\right) J_0(n\sqrt{t^2-a^2})dt,$$ where $Ai(t)$ is the Airy function and $J_0(t)$ is the zeroth order ...