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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0
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1answer
35 views

Are these integrals convergent?

Recently I've come across two integrals that seemed hard to check for me. Here they are: $$\int_{0}^{\infty} \frac{x \sin \ln x}{x^2 + \cos x} \, \mathrm{dx}$$ And another: $$\int_{1}^{\infty} ...
0
votes
2answers
71 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
35 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 ...
8
votes
3answers
137 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
vote
0answers
45 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
18 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 ...
0
votes
3answers
30 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 ...
0
votes
1answer
44 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
vote
0answers
26 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
vote
2answers
44 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: $(*) \, ...
3
votes
4answers
83 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
vote
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 ...
2
votes
2answers
43 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 = ...
2
votes
1answer
80 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} ...
2
votes
4answers
89 views

Prove that $\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}dx$ is convergent

Could you tell me how to prove that $$\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}dx$$ is convergent?
0
votes
1answer
36 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...
1
vote
0answers
75 views

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods and without gamma functions?

I know that $$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$ but all of the methods I've found seem to be too complicated for an early calculus student. Is there any method of calculating this ...
1
vote
1answer
58 views

Infinite integral of $1/(1+x^2)$

Given the theorem that the infinite integral of $1/x^n$ is convergent if and only if $n>1$, I want to prove that the infinite integral of $1/(1+x^2)$ exists. This seems like a trivial question, I ...
1
vote
2answers
38 views

Separation of integral by approximation

I'm working with the following integral $\displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}}$ and would like to split it in something like $$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x ...
3
votes
1answer
65 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
0
votes
1answer
45 views

How to solve this exp Integral

I am trying to solve the following integral, $$ I = \int_0^\infty \mathrm{e}^{z/2 - {\left(z - \ln a\right)^2}/4b} - \mathrm{e}^{z/2 - {\left(z + \ln a\right)^2}/4b}dz $$ where $a$ and $b$ are some ...
-4
votes
1answer
58 views

Convergence of $\int_0^1 \frac{\sin(x)}{x^2} dx$ [closed]

Can someone help me how to prove that $$\int_0^1 \frac{\sin(x)}{x^2} dx$$ does not converge?
3
votes
3answers
82 views

Prove that $\int_0^{\infty} \frac{\log (1+x)}{x^2}dx$ is divergent.

Could you please tell me how to prove that $$\int_0^{\infty} \frac{\log (1+x)}{x^2}dx$$ is divergent? I calculated an indefinite integral but I don't know how to prove that it diverges.
0
votes
1answer
83 views

Compute $\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}$

Could you tell me how to compute $$\int_0^{\infty} \frac{1}{\sqrt{x^3 + x}}dx$$ I have really no idea how to do this and I've tried for a couple of hours.
-1
votes
2answers
72 views

$\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$?

Is the following true or false: If $\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$? This should be doable without series.
6
votes
1answer
68 views

Find $\lim_{n \to +\infty} \int_{0}^{\infty} e^{-x} (nx - [nx]) dx $ [closed]

Find the limit $$\lim_{n \to +\infty} \int_{0}^{\infty} e^{-x} (nx - [nx]) dx $$ where $n$ is a natural number and $[nx]$ denotes the largest integer that is not greater than $nx$.
2
votes
3answers
120 views

Integral: $\int_{0}^{x}\lfloor\dfrac{1}{1-t}\rfloor dt$

I'm working on an integral problem(the rest of which is irrelevant) and this integral arises, which has stumped me. $$\int_{0}^{1}\int_{0}^{x}\left\lfloor\dfrac{1}{1-t}\right\rfloor dt dx$$ ...
1
vote
2answers
72 views

Convergence/divergence of a messy integral: $\int_1^\infty\frac{\arctan(9x)}{1+9x^3}\:dx$

Considering $$ \int_1^\infty\frac{\arctan(9x)}{1+9x^3}\:dx $$ I am trying to show convergence but looking to use Dirichlet's test and wanted to see if we can do it this way. Are we supposed to show ...
0
votes
0answers
17 views

Improper integral over “power exponential” function [closed]

The function exp(-(t/tempt)^beta) is used in medical research to fit gastric emptying curves normalized to 1 at t=0. It's called "power exponential" in this ...
3
votes
2answers
128 views

Closed form of $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor|\tan x|\rfloor}{|\tan x|}dx$?

I am trying to find a closed form for the integral $$I=\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor|\tan x|\rfloor}{|\tan x|}dx$$ So far, my reasoning is thus: write, by symmetry through ...
1
vote
2answers
72 views

Deadly integral

How to solve this question $\int\limits_0^1\frac{x^{2}+x+1}{x^{4}+x^{3}+x^{2}+x+1}dx$ . Please help me in solving this short way my approach is in the answer Is it correct and can it be solved in ...
0
votes
0answers
17 views

How to prove reflection positivity for $|x|^{-p}$ using Fourier transform (and contour integrals)

This question looks quite lengthy because I'm sketching the proof in the lecture - the two questions (look out for something bold) are actually relatively short. I need some help with a proof in our ...
-2
votes
1answer
83 views

A Beautiful Integral: $\int_{0}^{\pi/2}\log(\sin x)\log(\cos x)\,dx$ [closed]

I have to find the value of $$\int_{0}^{\frac{\pi}{2}}\log(\cos(x))\log(\sin(x))dx$$ in terms of $\pi$ and $\log(2)$. Any hint?
2
votes
1answer
35 views

Closed form for this integral (looks like Bessel)

I'm struggling to find a closed form for the following distribution (which is after all a Fourier Transform) written in integral form: $$I=\int_0^\infty\!\!\text{d}k\ \frac{ k }{\sqrt{k^2+m^2}}\sin(k ...
0
votes
3answers
48 views

express integrals as limits

How would you go about expressing the following as a limit? $$\int_0^1 \ln(x) dx$$ I know how to express limits on simple equations, but have no clue how to go about expressing an integral as a ...
0
votes
1answer
24 views

Convergence/divergence of $\int_0^1 \frac{\sin(\frac{1}{x})}{\sqrt{x}}dx$?

Does the following converge or diverge: $$\int_0^1 \frac{\sin(\frac{1}{x})}{\sqrt{x}}dx$$ I was thinking simply $$\int_0^1 \bigg |\frac{\sin(\frac{1}{x})}{\sqrt{x}}dx \bigg | \le \int_0^1 ...
0
votes
1answer
32 views

Evaluating a Erfc integral

I am trying to solve the following integral $$\int_0^{\infty } \int_{a-b x}^{\infty } \exp \left(-u^2\right) \, du \, dx.$$ I know it can be represented as an integral of the complementary error ...
2
votes
1answer
40 views

Why the sum of two divergent integrals has to be divergent?

Suppose $f(x)$ is a function defined on $\mathbb{R}\setminus\{c\}$, where $c$ is a scalar. Consider the integral $$\int_a^bf(x)dx,$$ where $a$ and $b$ are such that $a<c<b$. All Calculus books ...
0
votes
1answer
17 views

Determine the convergence of integral-Bound help needed

I have the following intergal: integral from 0 to infinity of (x^2)/(2x^3-x+1). I do not know how to create an inequality that will help me determine this convergence. Also I have a general question: ...
12
votes
3answers
852 views

Integral involving logarithm: $\int_0^\infty \frac{ \ln x}{(x+a)(x+b)} dx$

How to solve the following integral $$\int_{0}^{\infty} \frac{ \ln x}{(x+a)(x+b)} dx,$$ where $a,b>0$ and $a \neq b$. I was looking for some kind of substitution. However, I don't see an obvious ...
1
vote
0answers
76 views

How to calculate this integral containing a ArcTanh function?

I'm trying to calculate this integral : $$I(z,k,a)= \int_1^\infty t^2 \operatorname{ArcTanh} \left(\sqrt{\frac{t^2-1}{t^2}} \dfrac{k}{z}\right)\, e^{-a\,t} \, dt$$ Where : ...
3
votes
3answers
79 views

How to prove $ \int_{-\infty}^{+\infty}\left|x\right|^ke^{-(x-3)^2/2}dx $ is finite?

How to prove this integral as following is finite? $$ \int_{-\infty}^{+\infty}\left|x\right|^ke^{-(x-3)^2/2}dx $$ k is a positive integer
2
votes
0answers
48 views

Understanding principal value integral

I'm reading the original article on distance covariance (link), and throughout the article the author uses the following lemma: Can someone please explain what he actually means by "principal value ...
0
votes
0answers
24 views

Improper integral using residue calculus and contours [duplicate]

Can someone please show me how to solve this improper integral "using residue calculus and appropriate contours"? $$\int_0^{\infty} \frac{1}{x^4+1}dx$$
5
votes
1answer
39 views

Evaluate the improper integral with residues.

$$\int_0^{\infty} \frac{x^2+1}{x^4+1}dx$$ What i've found are the singularities at: $e^{\pi/4+\pi/2n}$ for $n=0,1,2,3$. But i'm unsure how to calculate the integral since I don't want to include ...
0
votes
0answers
104 views

On an estimation of a integral

I have the following function \begin{equation} S(x)=\int_{x_0}^x exp \left(-2 \int_{x_0}^y \frac{\beta(n-z)-a}{\beta(n-z)+a}dz \right)dy \end{equation} defined for $x \in [0, n+\frac{a}{b}]$ where ...
1
vote
1answer
78 views

Seeking help with an error function Integral

I am trying to compute the following Integral $$ I = \int_{0}^\infty x \exp \left(-2 x \right) \operatorname{erf}\left(\frac{x}{t^{H}\sqrt[4]{2}}-\frac{t^H}{2^{3/4}}\right) \, dx $$ where ...
1
vote
2answers
90 views

Can the limit of averages of $f(1),f(2),\dots, f(n)$ be expressed as an integral?

If $\int_0^1 f(x) dx$ exists then, of course, $$ \lim_{n\to\infty} \frac{f(\frac{1}{n})+f(\frac{2}{n})+\ldots+f(\frac{n}{n})}{n} = \int_0^1 f(x) dx. $$ I would like to know is there a similar formula ...
3
votes
0answers
84 views

On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
1
vote
1answer
34 views

On $-\frac{\zeta'(x)}{x\zeta(x)}$ and von Mangoldt function

I believe that it is possible show the following Fact. For real $x>e$ then $$-\frac{\zeta'(\log x)}{x\zeta(\log x)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{\log x}},$$ where $\zeta(x)$ is the ...