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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.

An improper integral is 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.

Specifically, an improper integral is a limit of the form:

$$\lim_{b\to \infty} \int_{a}^{b} f(x) \ dx \,,\ \lim_{a \to -\infty} \int_{a}^{b} f(x) \ dx$$ or of the form $$\lim_{c \to b^{-}} \int_{a}^{c} f(x) \ dx \,,\ \lim_{c \to a^{+}} \int_{c}^{b} f(x) \ dx$$

in which one takes a limit in one or the other (or sometimes both) endpoints.

Often, we can compute values for improper integrals, even when the function cannot be integrated in the conventional sense (as a Riemann integral, for instance), because of a singularity in the function or because one of the bounds of integration is infinite.

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Why does not $\int_{-\infty}^\infty x\,\mathrm{d}x$ converge?

It seems natural that it should converge, because for any $A\in\mathbb{R}$, $$\int_{-A}^A x\,\mathrm{d}x=\int_{-A}^0 x\,\mathrm{d}x+\int_0^A…
user197402
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Prove $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$

I try to evaluate following integral $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$ It seems okay to extend $\int_{-\infty}^{\infty}…
phy_math
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Which singularities are integrable

Consider a function $f: \mathbb{R}^n \to \mathbb{R}$ that has a singularity at point $x_0$ such that $\lim_{x\to x_0} f(x) = +\infty$. When we can be sure that this singularity is integrable? (you can assume that f is continuous for $x \neq x_0$ and…
D. Dmitriy
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Evaluating an integral via differentiation under the integral sign

I recently came across Feynman Integration here, thanks to one of Lucian's links I followed from a comment thread today. It has been quite an interesting read, I must say, but sadly I am (hopelessly) stuck on the concluding exercise. The first…
Guy
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Example of continuous positive function without limit whose improper integral is convergent

I would like an example of a function that is continuous and positive and has the following properties: $$\int_a^{\infty}f(x) dx $$ is convergent and $$\lim_{x \to \infty} f(x) \not = 0$$ (I think the limit should not exist).
user42768
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Compute $\int_0^\infty\frac{\cos(xt)}{1+t^2}dt$

Let $x \in \mathbb R$ Find a closed form of $\int_0^\infty\dfrac{\cos(xt)}{1+t^2}dt$ . Let me give some context: this is an exercise from an improper integrals course for undergraduates. My teacher was not able to give a valid solution without …
Gabriel Romon
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The reason that conditionally convergent integral produce different results

A well known result is that for \begin{equation} f(x,\,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}, \end{equation} one…
Yair
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Improper integral $\int_0^\infty\frac{\sin(\sin x)}xdx$

I know that the improper integral $\int_0^\infty\sin(\sin x)dx$ diverges since 2 ways of summing it give different values. Now my question is whether that divided by $x$ converges. I have a feeling that it does since it's similar to $\frac{\sin…
mini eden
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Does $ \int_0^{\infty} f(t) e^{-\alpha t} dt $ converge?

I've encountered this question in a physics textbook, it is not stated in the most accurate manner and the solution provided doesn't help me at all to understand it (paraphrasing the question): Suppose $f:[0, \infty) \to \Bbb{R}$ is a funciton such…
Gamow Drop
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Improper integral depending of one parameter

I would like to show that $$\int_0^{\infty}\frac{\sin x}{x}e^{-nx} \, dx=\arctan\frac{1}{n}$$ Any help is appreciate!
Barbara
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convergence test : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $

I have to check if $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $ is convergent or divergent. My approach was to integrate the function , hence : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx=-\lim_{x \to \infty} 1/\ln(x)+ \lim_{x \to…
sigmatau
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Computing $\int_{0}^{+\infty}\frac{\text{d}x}{1+xe^x}$

I've shown that the following integral exists $$ I=\int_{0}^{+\infty}\frac{\text{d}x}{1+xe^x} $$ WolframAlpha tells me that $I \approx 0,767$ but I can't find a way to compute the exact form. Any tips ?
Atmos
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Improper Integral $\int_{1/e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $

I need some advice on how to evaluate it. $$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $$ Thanks!
Ofir Attia
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How to calculate $\int^{\infty}_0 \frac{e^{-x^2}}{(x+\frac{1}{2})^2} dx$?

I am supposed to use the Euler-Possion-Integral $\int^{\infty}_0 e^{-x^2} dx = \frac{\pi}{2}.$ Here is what I've done so far: $$\begin{array}\ \int^{\infty}_0 \frac{e^{-x^2}}{(x+\frac{1}{2})^2} dx &= -\int^{\infty}_0…
Bloodpolyhydron
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Judge the convergence of the $\int_{0}^{\infty}\frac{x\ dx}{1+x^4\sin^2x}$

Judege if the improper integral is convergence $\int_{0}^{\infty}\frac{x\ dx}{1+x^4\sin^2x}$. My work: I want to change it into a series: the series=$\sum_{0}^{\infty}\int_{k\pi}^{(k+1)\pi}\frac{x\ dx}{1+x^4\sin^2x}$, and then use the…
fractal
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