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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1
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1answer
61 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
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2answers
44 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 \sqrt{...
3
votes
1answer
74 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
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1answer
49 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 ...
3
votes
3answers
88 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
85 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
76 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
132 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. $$\int_{0}^{1}\int_{0}^{x}\left\lfloor\dfrac{1}{1-t}\right\rfloor dt dx$$ $\bf\...
1
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2answers
78 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 ...
3
votes
2answers
135 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 $x=\pi/...
1
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2answers
81 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 ...
1
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0answers
20 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
95 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
42 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
51 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 limit....
0
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1answer
29 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 \frac{1}{\...
0
votes
1answer
38 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
47 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
869 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
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0answers
83 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 : $\operatorname{ArcTanh(...
3
votes
3answers
81 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
3
votes
1answer
88 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
26 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
43 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 the ...
0
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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
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1answer
81 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
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2answers
91 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
87 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
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1answer
36 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
77 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
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1answer
58 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 $\int_{0}^...
1
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1answer
39 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 45\...
2
votes
4answers
102 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
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2answers
63 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
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1answer
31 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
41 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 f'...
1
vote
1answer
36 views

Definite integral involving algebraic, exponential, and product of two Meijer's G function

I am having trouble with calculating the following integral: \begin{equation} I = \int_{0}^{\infty}x\exp({-\beta x})\large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} \...
0
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0answers
20 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
56 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
55 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
794 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
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0answers
14 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 $[f]\...
2
votes
0answers
47 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
37 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
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1answer
37 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
57 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
35 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
64 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 $$\int_0^\infty \exp(-sk)\sin(kx)\,dk=\frac{x}{x^2+ s^2}.$$ ...