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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2
votes
4answers
61 views

Convergence of $\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}$

I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?
1
vote
1answer
26 views

Proving that $\left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3$, given $1\leq a<b$

If $1\leq a<b$, then $$ \left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3.$$ Proceeding by integration by parts; let $u(x)=\sin(x)$ and $dv(x)=1/x$, then $u'=\cos(x)$ & $v(x)=\log(x)$. We ...
1
vote
1answer
27 views

Why is $\frac{1}{x^{1/p} (\ln(x)^2+1)}$ in $L^1$ but not in $L^p$ for any $p>1$

From a practice qualifying exam, the goal is to find a function $f \geq 0$ on $(0,\infty))$ that $f \in L^p(0,\infty)$ iff $p=1$. One function suggested was: $$\frac{1}{x^{1/p} (\ln(x)^2+1)}$$ So ...
0
votes
1answer
28 views

help with improper integral claim [duplicate]

We are finding difficulties in solving this claim: Let's suppose that $$ \int_a^\infty f(x)^2 dx < \infty \text{ and } \int_a^\infty f''(x)^2 dx < \infty. $$ Prove that $$\int_a^\infty ...
0
votes
0answers
15 views

Show $ \int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2} $ if $ a < b$

Show that if $a < b$: $$ \int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2} $$ I could see solution by Fourier analysis or by Contour Integration, but why does the larger frequency ...
2
votes
2answers
46 views

Complex - How to approach improper integral

I'm trying to solve this integral $$ \int_{-\infty}^{\infty} \frac{\sin(at) \sin(b(u-t))}{t(u-t)} dt $$ where $a$ and $b$ are positive. Any ideas how to approach this?
0
votes
1answer
45 views

A simple proof of the fact that $\int_0^{+\infty} \cos(x)/\sqrt{x} \text{d}x \neq 0$

When doing an exercise, I found that a sequence $(u_n)$ satisfies the following $$ u_n \underset{n\to + \infty}{\sim} \frac{1}{n^{\alpha/2}} \int_0^{n^\alpha} \frac{\cos(x)}{\sqrt{x}} \text{d}x, $$ ...
0
votes
0answers
25 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
0
votes
1answer
278 views

How to prove this integration is finite

I got stuck in verifying the condition that the following integration is finite: $$\int_{-\infty }^{\infty }\frac{\left | x \right |^\alpha }{\pi (1+x^2)}dx \tag{1}$$ The answer is that the ...
0
votes
0answers
10 views

Normed-Spaces and Integrals Question

Notations: $[f]$ is the equivalence class of $f$. $^\ast\int_{\mathbb{R}^n}f$ is the upper integral of $f$ $_\ast\int_{\mathbb{R}^n}f$ is the lower integral of $f$ Functionals ...
2
votes
0answers
36 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
1
vote
1answer
35 views

How to evaluate $\lim_{c \rightarrow \infty} \int_{-c}^c f(x)dx$

I'm trying to evaluate: $$\lim_{c \rightarrow \infty} \int_{-c}^c \frac{1+x}{1+x^2}dx$$ but I don't understand how to evaluate $$\lim_{c \rightarrow \infty} \int_{-c}^c f(x)dx$$ How?
19
votes
3answers
4k views

Evaluate $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x}dx$

How evaluate this integral? $$I=\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x}\,dx$$ Note: $$\int_0^{\pi/2}\frac{x^2\log{(\sin x)}}{\sin^2x}dx=\pi\ln{2}-\frac{\pi}{2}\ln^22-\frac{\pi^3}{12}.$$ ...
1
vote
1answer
33 views

What is the asymptotic behavior of this integral?

The function $F(x)$ is defined by the following integral $$F(x)=\int_0^x\frac{\left(1-y^3\right)^a}{\sqrt{\left(\dfrac{1-y^3}{1-x^3}\right)^b-\left(\dfrac{y}{x}\right)^4}}\,dy$$ where $a$ and $b$ ...
3
votes
0answers
30 views

How to prove $\lim_{a \to + \infty}a^q \int_{a}^{+\infty}\frac{\sin(x)dx}{x^p}=0$ when $p>q>0$

I know a similar problem in demidovich's problem set #2357 about proving $$\lim_{x \to 0^+}x^a\int_{x}^1 \frac{f(t)}{t^{a+1}}dt$$it proves by dividing the integral into two parts and used two ...
0
votes
2answers
51 views

Find $\int_0^{\infty} \frac{dx}{1+e^x}$

$$\int_1^\infty\frac{dx}{1+e^x} $$ $$\lim_{M\to\infty}\int_1^M\frac{e^xdx}{e^x(1+e^x)} \\ u= 1 + e^x \\ du = e^x dx \\ \lim_{M\to\infty} \int_{1+e}^{1+e^M} \frac{du}{(u-1)u} $$ I then found the ...
3
votes
3answers
270 views

Evaluating an improper integral that involves $\exp(-|x|)$

I am trying to prove that the function $f:\mathbb C\setminus\mathbb R\rightarrow\mathbb C$ defined by $$ f(z) := \frac{1}{2\pi i}\int_{-\infty}^\infty\frac{\exp(-|x|)}{x-z}dx $$ is holomorphic. I ...
100
votes
19answers
14k views
2
votes
3answers
681 views

Does $f(x)\in L^1$ imply that $\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0$?

Suppose that $f(x)$ is $L^1$ and R- integrable function, problem is to resolve if it is possible existence of such a $f(x)$ that: $$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ...
2
votes
2answers
102 views

How can $\int_a^b f(x)dx $ exist if either $f(a)$ or $f(b)$ does not exist?

In class, I came across the integral: $$\int_0^1 \frac{dx }{\sqrt{1-x^2}}=\frac{\pi}{2}$$ This is easy enough to prove using a substitution or by recalling the derivative of $\arcsin x$. However, ...
15
votes
1answer
644 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
1
vote
2answers
57 views

Evaluate the improper integral $\int_0^\infty {\exp(−sk)\over k}\sin(kx)\,dk$.

$$\int_0^\infty {\exp(−sk)\over k}\sin(kx)\,dk$$ I've tried hard for this but of no use.I've applied integration by parts by which I get $$\int_0^\infty \exp(-sk)\sin(kx)\,dk=\frac{x}{x^2+ s^2}.$$ ...
0
votes
2answers
33 views

Solving the improper integral $1/(x^a+y^b)$

I want to discuss the convergence of this improper integral: $$\int_{1}^{\infty }dy\int_{1}^{\infty }dx \frac{1}{x^\alpha +y^\beta} \text{ with } \alpha,\beta>0$$ I know by polar coordinates that ...
1
vote
1answer
18 views

How to find the inverse Fourier transfmation of exp(-sk)/k.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of exp(-sk) which is $$ \frac{\sqrt2}{\sqrt pi}\frac{x}{x^2+ s^2}$$ .After ...
1
vote
2answers
48 views

How to prove if $\int^{\infty}_{0}f(x)dx$ a converges, then there is increasing sequence $x_n$, $\lim_{n \to \infty}f(x_n)=0$

I tried to prove it directly, but examples like $\sin(x^{2})$ makes it impossible to find the proper subsequence $x_{n}$; I also tried proving by contraposition, but the converse negative statement ...
11
votes
2answers
333 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
0
votes
3answers
40 views

For what values of K, is the integral improper?

For what values of $K$ ($K > 0$), is the following integral improper? $$\int_{0}^{K}\frac{x}{x^2-2}$$ Now, I know that the function is undefined at $x=\sqrt{2}$. I also figured out that the ...
1
vote
1answer
61 views

Evaluate for $t\in \mathbb{R}$ $\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx$

Evaluate for $t\in \mathbb{R}$ $$\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx.$$ Here is what I have done: Let $f(z)={e^{itz}\over (1+z^2)^2}$. This has two poles $z=i$ $z=-i$ and an essential ...
2
votes
1answer
42 views

Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
3
votes
1answer
64 views

A clean way to obtain an (analytic or numeric) solution for this integral?

A friend and I have been looking at the crazy integral $$\iiiint \limits^{\infty}_{-\infty}\exp\left[-(x-t)^2-(x-h)^2-(y+t)^2-(y-h)^2-10\right]\mathrm{d}V$$ and can't come up with a decent method on ...
3
votes
2answers
45 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
4
votes
2answers
75 views

Fourier Transform leading to $\delta$: How does the Integration work?

So it is well-known that the complex exponential $$f(t) = e^{i\omega_0t}$$ has Fourier transform $$F(\omega) = 2\pi \delta(\omega-\omega_0) \ .$$ The transformation integral $$F(\omega) = ...
4
votes
4answers
122 views

Show that $\int_{-\infty}^\infty {{x^2-3x+2}\over {x^4+10x^2+9}}dx={5\pi\over 12}$

Show that $$\int_{-\infty}^\infty {{x^2-3x+2}\over {x^4+10x^2+9}}dx={5\pi\over 12}.$$ Any solutions or hints are greatly appreciated. I know I can rewrite the integral as $$\int_{-\infty}^\infty ...
3
votes
1answer
42 views

Using residue theorem to integrate from $-\infty$ to $\infty$

I'm trying to integrate $$\int_{-\infty}^{\infty} {x^2 \over {(x^2 + 1)}^2(x^2 + 2x + 2)} $$ given that the function $$f(z) = {z^2 \over {(z^2 + 1)}^2(z^2+2z+2)} $$ has residues $${9i - 12 \over ...
1
vote
0answers
40 views

What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
2
votes
2answers
46 views

Improper integral with module

faced with a problem when calculating the value of the integral $$ \int_{0}^{\infty} e^{-x}|\sin(x)|\, \mathrm{d}x$$ Is there any idea how?
4
votes
0answers
83 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
4
votes
1answer
48 views

$n$-th derivative of Beta function

We pretty much know nothing about the high order derivatives of the Beta function. Well, we known for the example some recursive formulae for $\Gamma^{(n)}(1)$ as well as ...
3
votes
4answers
89 views

Convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$

I want to test the convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$. There are some parts of the solution which does not make sense to me, I'm hoping that someone can explain ...
17
votes
4answers
7k views

Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.

$\sin x^2$ does not converge as $x \to \infty$, yet its integral from $0$ to $\infty$ does. I'm trying to understand why and would like some help in working towards a formal proof.
0
votes
1answer
28 views

Study the convergence of this improper integral

$$ \int_o^\infty t^ae^{bt}dt $$ for a,b reals. I guess I would have to separate this integral in many cases for different values of a and b. I know that if b < 0, $$ \int_o^\infty t^ae^{bt}dt ...
5
votes
1answer
43 views

Can someone help me with my proof about a limit evaluation?

Problem: Let $f:[0,1[ \to \mathbb{R}$ be a non-decreasing function such that $\int_0^1{f(x)dx}<+\infty $. Show that $$ \lim_{x\to 1^-}{(1-x)f(x)}=0.$$ Proof: $f(x)$ is a monotonic function so it ...
2
votes
0answers
49 views

Regularizing the sum of all factorials

Consider the series $$\sum_{n=0}^\infty n! = 0! + 1! + 2! + 3! + 4! + \ldots = 1 + 1 + 2 + 6 + 24 + \ldots$$ This series clearly diverges. Now, given that the Gamma function is defined by $$n! = ...
0
votes
3answers
42 views

Prove that a function is $L^p(\mathbb{R})$

There is a specific criterion for proving that a function $f \in L^p(\mathbb{R})$ as well as proving it by definition ? Furthermore, is correct to imply that: If $|\ f|^{\ p}$ is continuous in ...
0
votes
1answer
19 views

Definite Integral $\int_0^\infty\exp(-\sqrt{x^2+y^2})\left(\frac{1}{x^2+y^2}+\frac{1}{\sqrt{x^2+y^2}}\right)dx$ (mod. Bessel funs 2nd kind $K_n$?)

I'm trying to solve the definite integral $I_1=\int_0^\infty\exp(-\sqrt{x^2+y^2})\left(\frac{1}{x^2+y^2}+\frac{1}{\sqrt{x^2+y^2}}\right)dx,$ with $y>0$ and is obviously symmetric (so boundaries ...
1
vote
1answer
54 views

How to find all values for $\alpha$ and $\beta$ such that $\int _0^{\infty }f\left(x\right)$ converge [duplicate]

$f(x) = \begin{cases} x^{\alpha }\left(1-cos\left(1-x\right)\right)^{\beta } & \text{if $\;\;\;0<x<1$} \\ \frac{1}{x^{\alpha }+x^{\beta }} & \text{if $\;\;\;1\ge x$} \end{cases}$ I ...
3
votes
3answers
72 views

Does $\int _1 ^\infty\frac {f(x)} x\,dx$ converge or diverge?

Let $f(x)$ be continuous in $[1, \infty)$ and $\int_{1}^{\infty} f(x)\,dx$ converge. I need to prove or disprove this: $\int_{1}^{\infty}\frac{f(x)}{x}\,dx$ converge. I think this is true but I don't ...
1
vote
1answer
24 views

Is this improper integral $\int_e^\infty \frac{dt}{t^a \log^b (t)}$ convergent?

$\int_e^\infty \frac{dt}{t^a \log^b (t)} $ What I've done is that for $t > e$, $$\int_e^\infty \frac{dt}{t^a \log^b (t)} \le \int_e^\infty \frac{dt}{t^a } $$, which converges for $a > 1$. ...
2
votes
1answer
20 views

Improper convergence of $ cos(x)/{x^{1/2}} $

I have to evaluate the convergence of the improper integral $ \int_1^\infty \frac {cos(x)}{x^{1/2}}dx $. As the function is continuous on every $ [1, M] $, I can tell that this function is Riemann ...
3
votes
1answer
47 views

Convergence of Riemann sums for improper integrals

I was considering whether or not the limit of Riemann sums converges to the value of an improper integral on a bounded interval. This appears to be true in some cases when the sum avoids points where ...