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
37 views

An explanation of the integration

So, the integral is: $$\int_1^2\frac{x-2}{\sqrt{x-1}}dx$$ If I copied correctly from the board, the teacher said if x approaches 1+, the function approaches +$\infty$. What is the difference between ...
2
votes
1answer
34 views

How to solve this improper integral? [duplicate]

The problem is: If $f(x)\in C[0,+\infty)$, $\displaystyle\lim_{x\to+\infty}f(x)=k\in\mathbb R$, and $b>a>0$, prove: $$\int_{0}^{+\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-k]\ln(\frac ba)$$ My ...
1
vote
0answers
21 views

Is this Brownian Integral identity correct?

$$\int_0^1 B_t dt=\lim_{\omega \to\infty}{1 \over {\omega}}{\int_0^{\omega}{Y_0+}X_t dt}$$ Where $B_t$ is simple brownian motion, and $X_t$ is a discrete random variable that can be 1 or -1 with ...
0
votes
1answer
39 views

Is there another way than linearization?

$$I= \int {\sin^mx \cos^nx }dx$$ I need a Hint on doing this integral a Successive Partial Integration but it seems that the problem shows up when $ m = 2k $ and $ n = 2p$ where $p,m \in \mathbb{N}$. ...
2
votes
4answers
55 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
1
vote
2answers
71 views

Closed form of the integral $\int_0^1 \frac{x^n}{1+x}\, dx$

I am trying to evaluate the integral $$\int_0^1 \frac{x^n}{1+x}\, dx, \;\;\; n \in \mathbb{N}$$ in a closed form. I tried tackling it using Beta Form $\displaystyle \int_0^1 ...
0
votes
2answers
35 views

Convergence of the improper integral $\int_{0+}^{1-} \frac{\log x}{1-x} dx$

Let $0 < t_{1} \leq t_{2} < 1.$ Then $$\int_{t_{1}}^{t_{2}} \frac{\log x}{1-x} dx = \int_{1/t_{2}}^{1/t_{1}} \frac{\log u^{-1}}{1 - u^{-1}}(-u^{-2}) du = \int_{1/t_{2}}^{1/t_{1}} \frac{\log ...
4
votes
1answer
70 views

Prove that $ f:(a,b)\to\mathbb{R}$ is integrable iff $\lim_{\epsilon\to0} \int_{[a+\epsilon,b-\epsilon]}f$ exists

I want to solve the following: Let $ f:(a,b)\to\mathbb{R}$ continous such that $f(x)\ge 0 $ for all $x\in(a,b)$. Show that $f$ is integrable iff $\displaystyle \lim_{\epsilon\to0} ...
0
votes
0answers
29 views

Improper Integral patterns [on hold]

Integration is one of those techniques that only gets better over time, as one does more problem, it becomes obvious when to use which integration technique. When to use substitution, by parts, etc... ...
5
votes
3answers
110 views

Improper Integral of a periodic function converges

Given $f(x)$ is a periodic function and $\int_0^p{f(x)}dx=0$. Show $\int_1^\infty\frac{f(x)}{x}dx$ converges. 1) I know this integral can be broken into ...
0
votes
0answers
27 views

improper integral, convergent?

Fix a parameter $\theta \in (0,1/2]$. I am trying to figure out for which values of the parameter $\alpha\in \mathbb{R}$ this integral converges. $$\int_0^1 \frac{(1-(1-x)^{-\theta})^2}{x^\alpha} ...
7
votes
2answers
172 views

Why does $\int_0^{\infty}\frac{\ln (1+x)}{\ln^2 (x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?

I'd like to see the reason why $$\int_{0}^{\infty}\frac{\mathrm{ln}(1+x)}{\mathrm{ln}^2(x)+\pi^2}\frac{dx}{x^2}=\gamma$$ where $\gamma$ is the Euler-Mascheroni constant. I don't have any 'neat ...
1
vote
0answers
19 views

uniform convergence and improper integral(convolution)

I want to show that $$\frac{d}{dx}(f*g)=(\frac{d}{dx}f)*g$$ where $f(x)=\frac{1}{\sqrt{x}}e^{-\frac{1}{x}}$, $g(y)$ is continuous and bounded. the convolutions are improper integrals. I'm now here ...
7
votes
1answer
112 views

How to find the value of $I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$

How to find the value of $$I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$$ If we put $$I_2=\int_0^\infty\frac{\arctan^2({x})\log({1+x^2})}{\sqrt{x}(1+x^2)}dx$$ After long ...
0
votes
1answer
31 views

Substitution in integral, how shall I proceed

Say we have $\int_2^\infty \frac{1}{(\log n)^{\log n}}dn.$ Let $u=\log n.$ We have the boundaries become $u=\log 2$ and $u=\infty.$ How should I proceed with $dn.$ I have $du=\frac{1}{n}dn,$ hence, ...
0
votes
1answer
31 views

Transform an Integral bounds from -inf, inf to 0 to 1

Good day, If i have an integral from -infinity to infinity, how do I change the bounds/limits to 0 to 1? I don't want to give exact question since this is part of an assignment. I know how to figure ...
4
votes
0answers
37 views

Integral involving modified Bessel function of the second kind

I would like to calculate the closed-form expression for the following integral: $$ I = \int_{0}^{\infty} x^{M}\exp(-\frac{x}{a})K_{\nu}(b\sqrt{1+x})\mathrm{d}x,$$ where $M$, $a$, and $b$ are all ...
2
votes
1answer
83 views

A mysterious limit related to the integral $\int_{0}^{+\infty}\left(1-\frac{\tanh(ax)}{\tanh x}\right)\,dx$

I have to show that the following limit: $$ \lim_{a\rightarrow0}\Big[\sin(\pi a)\int_0^\infty \left(1-\frac{\tanh ax}{\tanh x}\right)dx\Big]=\pi\ln2$$ holds. This problem relates to my previous ...
1
vote
1answer
35 views

Showing the integral $\int_1^\infty \frac{1}{x(x+p)}\,dx$ is convergent for $p$ greater than $-1$.

Can someone help me why this is true: $$\int_1^\infty \frac{1}{x(x+p)}\,dx = \frac{1}{p}\int_1^\infty\left(\frac{1}{x}-\frac{1}{x+p}\right)dx$$
1
vote
2answers
25 views

Interchange of limit operator and $\ln$ function.

$$\lim_{n\to \infty}\ln\left(\frac{1+a^2n^2}{1+n^2}\right)$$ Can someone help evaluate that for me?
1
vote
1answer
43 views

Integration on manifolds and improper integration

Consider the usual concept of integral on a smooth manifold (the one built using partitions of unity). When applied to the usual smooth structure of $\mathbb{R}^n$, does it coincide with the concept ...
1
vote
2answers
36 views

Improper Integral of $xe^{-x}$.

I was working on this problem but I didn't get the right answer, though I can't find my mistake. Here is the question and my attempt: $\int_a^\infty xe^{-x}dx$ evaluate. $\lim_{b\to \infty} ...
-1
votes
2answers
89 views

How to solve the Integral $\int_0^{+\infty} \frac{1}{y}e^{-(y+\frac{x}{y})} \,dy$ [closed]

I have to find the integral: $$I=\int_0^{+\infty} \frac{1}{y}e^{-(y+\frac{x}{y})} \,dy$$ Where $x$ is a positive constant. I tried integration by parts but I could not get any thing. Any clue will ...
1
vote
0answers
15 views

Clarification of the idea of notations used in integral test proof

I'm looking through some notes presented on the Integral test proof and have been confused over the use of the notations and the concepts associated with the use of the notations like ...
7
votes
1answer
95 views

Proof of an integral identity involving $\pi$ and e

In the "Surprising Identities" post from a while back, Vladimir Reshetnikov offered the following identity[1]: $$\int_{0}^{\infty} dx \frac{1}{1 + x^2} \frac{1}{1 + x^{\pi}} = \int_{0}^{\infty}dx ...
1
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0answers
35 views

Holder continuity of the convolution of a Holder continuous function

Let $f(\theta, t)$ be a Holder continuous function for every $t$ on the interval $\theta \in (\alpha,\beta)$. It is known that the application of a singular operator to this function results in ...
0
votes
1answer
47 views

How do I show that $ \int_0^1 \frac{1}{x^3+\sqrt{x}}$ is convergent and that $\int_0^1 \frac{1}{x^5 + x}$ is not? [closed]

How do I show that $ \int_0^1 \frac{1}{x^3+\sqrt{x}}$ is convergent and that $\int_0^1 \frac{1}{x^5 + x}$ is not?
2
votes
1answer
28 views

Evaluate : $\lim_{k\to\infty}\int_0^\infty {1\over1+kx^{10}}dx$

Evaluate : $\displaystyle\lim_{k\to\infty}\int_0^\infty {1\over1+kx^{10}} \, dx$. From reading other answer to similar questions, I realized that I may have to use dominated convergence theorem to ...
9
votes
5answers
181 views

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

I have found myself faced with evaluating the following integral: $$\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx. $$ Mathematica gives a closed form of $8 \pi ^3/(81 \sqrt{3})$, but I have no ...
3
votes
3answers
69 views

Evaluating the improper integral $\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$

I am supposed to solve this integral but i have no idea how: $$\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$$ Since one limit is $0$ it will be divided by zero. Can someone please explain ...
6
votes
2answers
248 views

How to solve a hard integral?

How prove $ \displaystyle \int _{ 0 }^{ \infty }{ (1+x)\arctan { (x) } } \log^4 { (x) }{\frac{1}{\sqrt{x}(1+x^2)}} dx=\frac{57\pi^6\sqrt{2}}{64} $ I found this integral using numerical values.I ...
1
vote
1answer
43 views

Integration of this using a multi-dimensional hypergeometric function

I want to try and potentially use a Dirichlet - Hypergeometric Function in order to compute the following integral. I would appreciate some help as I'm stuck on how to go about this is a ...
0
votes
1answer
28 views

Determine whether it is convergent or divergent: $\int_{-1}^0 {\frac{e^{1/x}}{x^3}}dx$

So I was evaluating this improper integral, and found the antiderivative to be $e^{1/x}(1-\frac{1}{x})$. How would I evaluate it from $0$ to $-1$? In other words, what would $\frac{1}{0}$ be? ...
1
vote
2answers
90 views

Various evalutions of $\int_0^\infty \sin x \sin \sqrt{x} \,dx$

I'm looking for various ways to evaluate the integral: $$\int_0^\infty \sin x\sin \sqrt{x}\,dx$$ I'm mainly interested in complex analysis. I can think of a wedge -shaped contour of angle $\pi/4$ but ...
1
vote
0answers
21 views

Question on Newman's proof of the Prime Number Theorem

I am reading through Zagier's exposition of Newman's proof of the prime number theorem and I do not understand one of his arguments when proving his so called Analytic Theorem. This theorem states the ...
0
votes
2answers
37 views

Proof of certain Gaussian integral form

I am having trouble understanding where the following integral form comes from: $$\int_{-\infty}^{\infty} e^{-a x^2 }e^{-bx}=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{a}}$$ I see and understand that the value ...
0
votes
2answers
40 views

Convergence of improper integral $\int_0^1 \frac{x^\alpha}{x+x^2}dx$ for $\alpha>0$

I'm having trouble showing the convergence of the integral in the title, for $\alpha >0$: $$\int_0^1 \frac{x^\alpha}{x+x^2}dx $$ I tried using: $$\int_0^1 \frac{x^\alpha}{x+x^2}dx\leq \int_0^1 ...
0
votes
1answer
32 views

Proving the transform of the Q-function

I have the Gaussian Q-function, given by: and I want to prove that it can be also expressed as: Can somebody help explaining how to obtain the second integral from the first?
1
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0answers
45 views

Relation between Nuttal Q-function and Gaussian Q-function

I am trying to express the famous Nuttal Q-function, given as: $$\mathcal Q_{m,n}(p,q)=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt$$ where $m$, $n$, $p$, and $q$ are constants and ...
0
votes
1answer
25 views

finding the free energy of a van der waals gas (integration)

I have the following integral, $\int{ \frac{-nrtV}{(v-nb)^{2}} dV}$ could anyone tell me how to do this?
0
votes
2answers
45 views

Convergence of $\int_0^\infty \frac{\sqrt{x}\sin x}{(e^x-1)\log(1+x)}dx$

Could someone please help me determine wether the following integral converges: $$\int_0^\infty \frac{\sqrt{x}\sin x}{(e^x-1)\log(1+x)}dx$$ I have no idea how to start unfortunately... So any hint ...
1
vote
0answers
22 views

Improper and definitive integral of trigonometric functions involving absolute values

Let $x(t)=10\cos(100t+300°)-5\sin(220t - 50°)$ . It is asked to evaluate the following integrals: $$\int_{-\infty}^\infty |x(t)|^2 dt \text{ and } \frac{1}{T} \int_{-T}^T |x(t)|^2 dt$$ Where $ T$ is ...
1
vote
2answers
30 views

A general formula for a specific improper integral

The integral I'm after is here: The question is a little ambiguous whether it wants a general solution for this, but based on thought, I would guess there are many different answers based on the ...
1
vote
5answers
149 views

(another) Challenging improper integral [closed]

$$\int_0^\infty y^{1/2}e^{-y^3}\,dy$$ It is in the section with the gamma function if that helps. Thanks!
2
votes
3answers
57 views

A challenging improper integral

The integral is $$\int_0^1\frac{dx}{\sqrt{-\ln x}}.$$ Not sure if it helps, but it is in the same problem section as $$\int_0^\infty e^{-x^2}dx.$$
0
votes
1answer
44 views

Improper integral show convergence/divergence

How do I show the convergence/divergence of this improper integral? $$\int_1^{\infty}\frac{3-x-x^2\sin x}{3+x+x^3}\,\mathrm dx$$
2
votes
1answer
60 views

How can i solve $\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau$,

I want to find the value of the integral $$\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau,$$ where $0<\alpha<1$. Using Mathematica I found the solution to be ...
1
vote
3answers
53 views

integral from 1 to infinity of $\frac{5}{(4x+2)^3}$

I have solved the integral: $$\int_1^\infty{\frac{5}{(4x+2)^3}}dx$$ using u substitution and I am not getting the correct answer. I am missing some step here or making an algebra error. I am not ...
2
votes
1answer
29 views

Functions such that $\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$ and …

Can we find a bounded function $a:\mathbb{R}\to\mathbb{R}$ and a function $f\in L^1(\mathbb{R})$ with $f\neq 0$ such that $$\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$$ and ...
1
vote
1answer
60 views

$\mu$ test for convergence of improper integral of first kind

While going through an Indian text on Analysis I found a test for convergence of improper integral.It was stated without proof.I tried to prove it..then some doubts pop up... Statement is this :Let ...