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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1answer
31 views

Integrals of a function and its absolute value

Is the following proposition true? Let $f(x)$ be a real-valued function defined on $[a,b] \subset \mathbb{R}$, and suppose that the integral, $$ I = \int_a^b f(x) dx, $$ exists in the sense of ...
4
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1answer
97 views

Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $

Given $$\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$$ evaluate: $$\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx. $$ I can find that $$\left(ax+\frac{b}{x}\right)^2 = a^2x^2+2ab+\frac{b^2}{x^2}...
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1answer
39 views

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, $$\int _1^{\infty }\frac{dx}{x\log ^2x} $$
1
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1answer
53 views

$w(x,y)=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dx - \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dy$ , calculate $\int_{+\gamma} w $

$\gamma$ is the curve of this equation: $$\rho=e^{-\theta} \qquad \theta \in [0,+\infty)$$ It is oriented in the growing $\theta$ $$w(x)= \sum_{i=1}^n a_i(x) \ dx_i $$ $$\int_{+\gamma} w=\sum_{i=1}^...
6
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0answers
189 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in ...
7
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3answers
181 views

Showing $\pi\int_{0}^{\infty}[1+\cosh(x\pi)]^{-n}dx={(2n-2)!!\over (2n-1)!!}\cdot{2\over 2^n}$

Showing $$\pi\int_{0}^{\infty}[1+\cosh(x\pi)]^{-n}dx={(2n-2)!!\over (2n-1)!!}\cdot{2\over 2^n}\tag1$$ Recall $$1+\cosh(x\pi)={(e^{x\pi}+1)^2\over 2e^{x\pi}}\tag2$$ $$I_n=2^n\pi\int_{0}^{\...
5
votes
3answers
204 views

Prove $\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}dx=\ln\left({\pi\over 2}\right)$

Integrate $$I=\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}\,dx=\ln\left({\pi\over 2}\right)\tag1,$$ where $\phi={1+\sqrt5\over 2}$. Recall $\tanh y=-{1-e^{2y}\over 1+e^{2y}}...
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5answers
2k views

Evaluate $\int_0^\infty\frac{1-e^{-x}(1+x )}{x(e^{x}-1)(e^{x}+e^{-x})}dx$

\begin{equation} \int_0^\infty\frac{1-e^{-x}(1+x )}{x(e^{x}-1)(e^{x}+e^{-x})}dx \end{equation} My colleague got this problem from his friend but he didn't know the answer so he asked my help. ...
3
votes
1answer
164 views

Contour integral for finding $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx$

I can't prove the following result: $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx=\frac{\ln \sqrt{a^2+b^2}}{b}\arctan\frac{b}{a}$ for all $a,b \in \mathbb{R}.$ Well, I consider $\...
2
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1answer
90 views

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'...
1
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1answer
51 views

Improper Integral of $f_n$ of a Uniformly Convergent Sequence

Let $(f_n)$ be a sequence of functions defined in $[a,\infty)$, which uniformly converges to $f$ in every interval $I_b$ of the form $[a,b]$. Assume every function in the sequence is integrable in $...
1
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1answer
99 views

Convergence and value of the integral $\int_0^\infty\ln\left(\frac{1+x^{(\sqrt3+2)}}{1-x^{(\sqrt3-2)}}\right) \frac{1}{(1+x^2)\ln x} \, dx$

Check the convergence of the following integral and, if it converges, compute its value: $$\int_0^\infty \frac{\ln\left(\frac{1+x^{(\sqrt3+2)}}{1-x^{(\sqrt3-2)}}\right)}{(1+x^2)\ln x} \, dx$$ ...
4
votes
4answers
123 views

Evaluating $\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$ using contour integration

I need your help with this integral: $$\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$$ where $a>0$. I have tried some complex integration methods, but none seems adequate for this ...
2
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1answer
41 views

Integral representation of a Meijer G-function

How to prove that, the integral $$I_{a,b}:=\int_{1}^{+\infty}e^{-at}(1-t^{-1})^b\,dt ; \, a,b>0$$ is given by $\Gamma(b+1)$ times a Meijer G-function, i.e., $$I_{a,b}:=\Gamma(b+1) \times G^{m,...
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3answers
83 views

principal value of $\int_{-\infty}^{\infty}\frac{\sin^2(x)}{x^2}\mathrm{d}x$

I know the answer is $\pi$ there is a proof here. Now looking to my textbook (textbook image) the result should be $0$. Using the last equation on the right hand page we have: $$ i\pi(\sin^2(x))'|_{x=...
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1answer
84 views

Evaluate the improper integral $\int _1^\infty x^{-10/9} \coth(x) \,\mathrm dx$

Could someone help me to evaluate this integral please: $$\int _1^\infty \dfrac{1}{x^{10/9} \tanh(x)} \,\mathrm dx$$ I tried using change variable method in order to change the integral bound.
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3answers
26 views

Double integral, positive and negative infinity as bounds

I'm not really sure how to approach this problem, at first I tried substituting a for positive infinity and b for negative infinity but got stuck when I tried to substitute these in after integrating ...
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1answer
26 views

Integral with exponential function

I'm trying to determine the order of an integral involving the derivative of the heat kernel on the real line. Based on some numerics, it appears as though the following identity holds: $$ \int_{\...
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0answers
35 views

Multi integral change of variables?

Sketch the domain D bounded by $y=x^2$, $y=1/2x^2$, and $y=2x$. Use a change of variables with the map $x=uv$, $y=u^2$ to calculate: $∫∫y^{-1}dxdy$ Ok so I found the Jacobian to be $-2u$, and ...
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2answers
53 views

Using ML inequality

I'm trying to work out the following integral: $$\int_{-\infty}^\infty \frac{1}{\cosh(x)} dx $$ by complex analysis methods and as a result of the contour I chose (a rectangle) one of the things I ...
0
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1answer
52 views

Understanding an apparent contradiction from naively applying Taylor's theorem and Fubini's theorem together

Suppose $f(x)$ is a bounded, $\mathcal{C}^{\infty}$ function on $\mathbb{R}$ for which the integral $$I = \int_{0}^{\infty}f(x)\ dx,$$ exists. Taylor's theorem implies $f$ admits a MacLaurin expansion ...
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0answers
29 views

On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
2
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0answers
43 views

Improper integral $\int_{-\infty}^{\infty} \frac{e^{sx}}{1+e^{x}} dx$

I'm having trouble solving this problem and was hoping someone could show me how it's done? $$\int_{-\infty}^{\infty} \frac{e^{sx}}{1+e^{x}} dx$$ where $0<Re(s)<1$
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1answer
26 views

Doubts concerning to an application of Frullani's theorem to $f_k(x)=\frac{2^{-k^2x}-2^{-(k^2+1)x}}{x}$, and Lebesgue convergence theorems

By application of Frullani's theorem for $a_n=n^2+1$, $b_n=n^2$ where $n\geq 2$ and $f(x)=2^{-x}$ then RHS in Frullani's integral is obtained for $n\geq 2$ as $$\log(1-\frac{1}{n^2}),$$ thus I asked ...
16
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3answers
359 views

Prove $\pi^2\int_0^\infty\frac{x\sin^4\pi x}{\cos\pi x+\cosh\pi x}dx=e^2\int_0^\infty\frac{x\sin^4ex}{\cos ex+\cosh ex}dx=\frac{176}{225}$

Marco Cantarini and Jack D'Aurizio proved hard-looking integrals (see Marco and Jack) in my recent two posts. This is our final hard-looking integral that yield a rational answer: $$\pi^2\int_{...
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1answer
22 views

Show that $\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$ is convergent for $\Re s > -1$

I am struggling to understand the example in Special Functions p. 621, that states, that $$\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$$ is ...
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0answers
18 views

Proof: Majorant- and minorantcriterion for convergence of improper integrals

Let $I$ a interval and let $f,g:I\to \mathbb{R}$ continuous with $0\le g(x) \le f(x) \forall x \in I$. Prove this propostitions: (a) If $\int_{0}^{\infty} f(x)dx$ convergent $\Rightarrow$ $\int_{0}^{\...
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1answer
55 views

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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0answers
44 views

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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4answers
126 views

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?
6
votes
1answer
106 views

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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3answers
33 views

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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1answer
36 views

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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3answers
139 views

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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1answer
61 views

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 ...
0
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1answer
50 views

Are these integrals convergent?

Recently I've come across two integrals that seemed hard to check for me. Here they are: $$\int_0^\infty \frac{x \sin \ln x}{x^2 + \cos x} \, \mathrm{d}x$$ And another: $$\int_1^\infty \frac{\sin \ln ...
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0answers
53 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
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2answers
82 views

Is $\frac{1}{x^2}$ Lebesgue integrable while $\frac{1}{x}$ is not?

My textbook defined integrability as $f$ is said to be Lebesgue integrable if $\int{}f$ is finite. I heard that $\frac1x$ is not Lebesgue integrable, but $\frac{1}{x^2}$ is Lebesgue integrable. I do ...
2
votes
2answers
41 views

Finding limit with improper integral

How should I approach this question? $$\lim_{x\to0}\frac{1}{x}\int_1^{1+x}\frac{\cos t}{t} \, dt$$ I tried to use L'hospital and that gave me $-\sin(0) = 0$ The correct answer is $\cos 1$. Did I L'...
8
votes
3answers
155 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
1
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0answers
63 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
0
votes
1answer
19 views

“Nonlinear cosine” integral

Let $\alpha > 1$, $\xi \in\mathbb{R}$. and $\chi_A$ be the characteristic function of the set $A$. Are there some known ways of computing (or estimating in terms of $\xi$) of this kind of ...
3
votes
3answers
116 views

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$ for $a, b > 0$. There are 2 problems. $|\sin(x)|^b = 0$ for $x = k \pi$ and $x^a = 0$ for $x = 0$. We can write $\int_0^{\...
0
votes
1answer
48 views

Having trouble evaluting error function integrals

I am trying to evaluate $$I = \int_1^{\infty } \left(\frac{\operatorname{erf}\left(a -b\log (x)\right)}{2 x^2}-\frac{\operatorname{erf}\left(a + b\log (x)\right)}{2 x}\right) \, dx$$ Let $\log (x) = ...
1
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0answers
44 views

Limit of improper integrals of uniformly convergent function

I've got a problem. Let $g(t)\ge0$ and it has improper integral on interval $[A, B)$. Furthermore, sequence of integrable functions $f_{n}(t)$ is uniformly convergent do $f(t)$ on every subinterval $...
1
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2answers
57 views

Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, \,\al(t)...
3
votes
4answers
94 views

Limit of $ \frac1x \int_x ^{2x}e^{-t^2}dt$

What is the limit of the function $$\lim_{x\to 0} \ \frac1x \int_x ^{2x}e^{-t^2}dt$$ ? I tried this problem by using gamma function. I couldn't find the integral.
1
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1answer
32 views

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_{...
2
votes
2answers
46 views

Convergence and value of improper integral

I have to prove that integral $I = \int_{0}^{+\infty}\sin(t^2)dt$ is convergent. Could you tell me if it's ok? Let $t^2=u$ then $dt=\frac{du}{2\sqrt{u}}$ Now $$I = \int_{0}^{+\infty}\frac{\sin(u)du}...
2
votes
1answer
83 views

Help evaluate $\int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx$.

I am trying to evaluate $$ I = \int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx $$ where $a \ge 0$ and $b> 0$. $$ I = \frac{2}{\sqrt{\pi}}\int_0^\infty \int_{a + b\ln (x)}^{\infty} \mathrm{e}^...