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
25 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
33 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
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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
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3answers
853 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
80 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
50 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
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0answers
25 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
40 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 ...
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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
79 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
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0answers
85 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 ...
0
votes
2answers
76 views

Arriving at the asymptotic $\int \limits_\lambda^\infty e^{-t^2/2}dt \sim \frac{e^{-\lambda^2/2}}{\lambda}$

In the book "The Probabilistic Method", the integral $\int_\lambda^\infty e^{-t^2/2}dt$ is said to be "approximately equal" to $\frac{e^{-\lambda^2/2}}{\lambda}$ for large $\lambda$. I assume what is ...
2
votes
1answer
52 views

Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality $\int_{0}^{\infty}e^{-tx}\,dx=\frac{1}{t}$

I'm learning about measure theory, specifically Lebesgue integral, and need help with the following problem: Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality ...
1
vote
1answer
38 views

How do I solve this indefinite integral?

Given the improper integral: $$\int_1^\infty 45\frac{x+1}{x^2+2x} \, dx$$ I was able to set up the limits as shown below, but I am not sure how to continue integrating. $$\lim_{t\to\infty}\int_1^t ...
2
votes
4answers
94 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
2answers
60 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
29 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
36 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
49 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
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1answer
51 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 $$ u_n \underset{n\to + \infty}{\sim} \frac{1}{n^{\alpha/2}} \int_0^{n^\alpha} \frac{\cos(x)}{\sqrt{x}} \text{d}x, $$ ...
10
votes
0answers
785 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
0answers
11 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
46 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
36 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?
1
vote
1answer
36 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 $b$ ...
0
votes
2answers
53 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
0answers
33 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 ...
1
vote
2answers
59 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 ...
2
votes
2answers
103 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, ...
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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 ...
0
votes
3answers
41 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
22 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 $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
2
votes
1answer
46 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 ...
1
vote
1answer
63 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 ...
3
votes
2answers
49 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
4answers
128 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
44 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
42 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
48 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
1answer
50 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
94 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 ...
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 ...
3
votes
1answer
68 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 ...
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 ...