# Tagged Questions

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 ∞ or −∞, or as both endpoints approach limits.

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### Quick question about improper integral

What do I do if in the point of lower bound of some first-odered improper intagral integrand doesn't exists? For instance, $$\int _1^{\infty }\frac{dx}{x\log ^2x}$$
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### How to evaluate $\int_{0}^{\infty}\arctan (x^2)\sin(x^2)\mathrm dx$

How to evaluate $$I=\int_{0}^{\infty}\arctan (x^2)\sin(x^2)\mathrm dx$$ with the help of Wolfram alpha,I got the answer below $$I=\frac{\pi^{2/3}\text{erfc(1)}(\text{erfi(1)}+1)}{4\sqrt2}$$ But I don'...
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### A closed form of a integral with exp and cos

Can we find a closed form for the following integral: $$\int_0^{\infty} \frac{e^{-x} \cos x}{1+x} \, {\rm d}x$$ No matter how hard I tried I cannot tackle it. I am pretty much afraid that if a ...
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### Evaluating a limit of an integral

I have a function $f(x,y,z) :\mathbb{R}^3 \rightarrow \mathbb{C}$, a smooth function. I know that $$I = \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} f(x,y,z) \ dx dydz$$ ...
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### Prove that $\int_0^1 \frac{\log{x}}{1-x^2}dx$ is convergent [closed]

Could you please help me with proving that $$\int_0^1 \frac{\log{x}}{1-x^2}dx$$ is convergent?
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### How was the difference of the Fransén–Robinson constant and Euler's number found?

I recently ran across the following integral: $$\int_{0}^{\infty}\frac{1}{\Gamma(x)}dx$$ Which I learned is equal to the Fransén-Robinson constant. On the linked wikipedia page for the Fransén-...
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### Showing the integral $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$ converges

I am trying to bound the following integral: $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$. I am very sure this integral converges, but whatever I try seem to ...
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### Integral convergence $\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}}$

I've tried to partial this integral from 0 to 1, 1 to e, and e to infinity. $$\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}} dx$$
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### Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$

I can't solve the integral $$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$ I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.
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### Does the integral converge

How to prove that the following integral doesn't converge? $$\int_0^\infty \frac{1}{(\ln^4x + \ln^2x)\ln^2(1-x^{1/3})^2(x + \sqrt{x} + 1)}dx$$ I suppose it doesn't converge because of quick growth ...
### Checking whther the integral $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ convergent
I need to check convergence of $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ . I think it divergence cause it bigger than $\int_1^∞ \frac{1}{x} dx$ but I can't prove it. I have an hint that \$\lim\limits_{...