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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6answers
54 views

Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$

I am familiar with the gauusian integral $$\int_{-\infty}^\infty e^{-\alpha x^2+\beta x}dx=\sqrt{\frac{\pi}{\alpha}}e^{\beta^2/(4\alpha)}$$ Could anyone help me to find out the value of the following? ...
1
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2answers
46 views

To test convergence of improper integral $\int_{0}^{1} \left(\log\left(\frac{1}{x}\right)\right)^m\,\mathrm dx$

To test convergence of improper integral $$\int_{0}^{1} \left(\log\left(\frac{1}{x}\right)\right)^m\,\mathrm dx$$ I made cases and I am stuck on case in which I have to check convergence for ...
1
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0answers
35 views

Convergence of improper integral $\int_{2}^{\infty} \frac{1}{log(t)}dt$ [duplicate]

Convergence of improper integral $\int_{2}^{\infty} \frac{1}{log(t)}dt$ How do i start?
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0answers
16 views

Testing convergence of a sum with two improper integral in it

For wich $\alpha$ does the series converge?$$\sum_{n=3}^{\infty}\sin{\{2\pi n^2 + [\int_{(\ln n)^\alpha}^\infty arctan(t)*(\sin {1/t})^3 dt]*[\int_{\ln n}^\infty \frac{arctan(t^2)}{e^t +2}dt]\}}$$ i ...
0
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2answers
56 views

To test convergence of improper integral $ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\, \mathrm dx$

I have to test convergence of improper integral $$ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\,\mathrm dx$$ I write as $\log(x) \leq x$ . So $x\log(x) \leq x^2$. So $ \frac{x\log(x)}{(1+x^2)^2} ...
1
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2answers
71 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
2
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3answers
93 views

Evaluating $\int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$

The problem is to evaluate the improper integral: $I = \int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$. This can be written as follows: $$I = \int_0^{\infty} dt \frac{\sin(xt)}t \int_0^a ...
5
votes
2answers
143 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
2
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1answer
71 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
2
votes
0answers
48 views

Sine improper integral

Suppose the following integral $$ \int\limits_{-\infty}^{\infty}\sin{x}dx $$ In mathematical rigor, the following is the definition $$ \int\limits_{-\infty}^{\infty}\sin{x}dx = ...
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1answer
52 views

How do I evaluate $\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\lambda}} dt$ ??

How do you evaluate: $$\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\alpha}} dt = ??$$ Many thanks.
2
votes
2answers
50 views

How to show the convergence of the integral $\int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t}t^x\mathrm{d}t$?

The integral is defined, for all $x\in\mathbb{R}$ as follows: $$I= \int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t} t^x\mathrm{d}t.$$ When $I$ converges? Let $t-1=u$, we have: $u\to 0$ when $t\to 1$. ...
0
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0answers
27 views

Convergence of improper integral with parameter

In my assignment I have to study the convergence of this integral: $$\int_{0}^{1} \frac{ln(1 + \sqrt{x})}{x (x^{\alpha}-1)} dx$$ with the parameter $\alpha >0$. In a neighbourhood of $x=0$ I ...
2
votes
1answer
32 views

Solve the improper integral: $\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$

I'm completely stuck on this one. I only know that it converges thanks to Wolfram, but I don't know how to evaluate it. $$\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$$ Thank you for the help.
1
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1answer
21 views

When to stop simplifying an improper integral

When evaluating whether an improper integral is convergent or divergent, I'm sometimes unsure whether I simplified enough to be sure. For example, given $$ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sec ...
2
votes
2answers
67 views

Compute integral: $\int_0^{+\infty}\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-\frac{1}{2} (x^2+y^2)}dx dy $

I would like to resolve this exercise: Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$. I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ ...
3
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1answer
52 views

Evaluating $\int_0^{\infty} \frac{\sin xt \sin yt \cos zt}{t^2} \, dt$

The problem is to evaluate the improper integral $I = \int_0^{\infty} \frac{\sin xt \sin yt \cos zt}{t^2} dt$. This can be written as $\int_0^{\infty} dt \int_0^y \frac{\sin xt \cos st \cos zt}{t} ...
3
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0answers
50 views

How can revolving an infinite area have a finite volume [duplicate]

The area of the region bounded by $f(x) = \frac{1}{x}$, $y = 0$, and $x = 1$ is $$ A = \int_1^{+\infty} f(x) \, \textrm{d}x = \lim_{b \to +\infty} \int_1^b \frac{\textrm{d}x}{x} = \lim_{b \to ...
1
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1answer
64 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
0
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2answers
40 views

Elementary integral for square roots of trig functions?

What's an easy way to calculate something like $\int \sqrt{1+\cos x} \text{ d}x$?
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2answers
55 views

Find $\int_0^\infty \frac{\sin(4x)}{x}$

How would one go about computing $$\int_0^\infty \frac{\sin(4x)}{x}$$ without any background in complex analysis (e.g. using strictly calculus)? I know that $$\int_0^\infty \frac{\sin(x)}{x} = ...
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0answers
34 views

Finding an integral for an expression

I have the equation below. If I work backwards and integrate the second line w.r.t. t and then evaluate at t = x, I can get the first line. However, how do I go from the first expression to the ...
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1answer
34 views

Generalisation of an already generalised integral

Inspired by these two questions: Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ Interesting integral formula I ask whether the following integral has a closed form: ...
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0answers
29 views

double integral with Gamma function

How can I solve the integral $$\int_{0}^{\infty}\int_{\delta t}^{\infty} e^{-x}x^{\beta-1}dxd\delta$$ or how can I check weather the integral is proper or not. Can anyone help me to reduce it into ...
1
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2answers
41 views

For which real $a$ is the integral $\int_1^\infty x^ae^{-x^3\sin^2x}dx$ finite?

For which $a \in \mathbb{R}$ is the integral $\int_1^\infty x^ae^{-x^3\sin^2x}dx$ finite? I've been struggling with this question. Obviously when $a<-1$ the integral converges, but I have no idea ...
0
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0answers
18 views

Solutions to Heat equation $ \int_{\mathbb{-\infty}}^{\infty} \frac{\partial^2 T(x, t)}{dt^2} dx = 0 $

I was wondering about the motion of heat and came across this differential equation. $$ \int_{\mathbb{-\infty}}^{\infty} \frac{\partial^2 T(x, t)}{dt^2} dx = 0 $$ $T(x,t)$ represents temperature ...
3
votes
3answers
61 views

Show that $\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}}$ converges.

Show that $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} $$ converges. I recognized that that since the integrand is even then $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} = ...
2
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1answer
49 views

What is the nature of this improper integral?

Consider this improper integral of first kind: $$\int_0^{+\infty}{\frac{t\ln t}{{(t^2+1)}^{\alpha}}}\,{dt}, \quad \alpha\in\mathbb R$$ Its required to find the nature of this improper integral. We ...
1
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4answers
87 views

How to solve this integral by parts?

I was solving a problem of mean values, and I would like to solve and evaluate this integral: $$ \langle x^2\rangle=\int_{-\infty}^{\infty}\left(\frac{2\alpha}{\pi}\right)^{1/2}x^2 ...
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3answers
53 views

Evaluate $\int_{2}^{+\infty} \frac{1}{x \log^2{x}} dx$

I'm having some difficulties solving this improper integral: $$\int_{2}^{+\infty} \frac{1}{x \log^2{x}} dx.$$ Taking the limit as $b$ approaches infinity we have $$\lim_{b\to\infty}\int_{2}^{b} ...
0
votes
1answer
49 views

Why does $ \int _1 ^3 \frac x {(x^2-9)^{4/3}}dx$ diverge?

Could someone please explain to me why the following integral is diverging? And how you would go about proving that it is. $$ \int_1^3 \frac{x}{(x^2-9)^{4/3}}dx $$
2
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1answer
48 views

Convergence of the integral $\int \limits ^\infty _ 0 \frac{dx}{\sqrt{x}(x+\cos(x))}$

My question is how to prove the convergence of integral $$\int \limits ^\infty _ 0 \frac{dx}{\sqrt{x}(x+\cos(x))}$$ I already have, that $\int \limits ^\infty _ 1 \frac{dx}{\sqrt{x}(x+\cos(x))}$ ...
2
votes
1answer
36 views

Twist on the integral test

I have a question regarding a slight modification of the integral test. Suppose that $a_k = f(k)$ for some continuous function $f:[1,\infty) \rightarrow [0,\infty)$, which satisfies $f(k)\rightarrow ...
1
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2answers
40 views

Finding a constant where an improper integral converges

I am trying to find the interval for the value $p$ where the following integral converges (log is the natural logarithm): $$\int_1^{\infty} \frac{\log{x}}{x^p} \,dx$$ So far I've used integration ...
0
votes
3answers
32 views

When does the improper integral of the sum of two functions converge?

Suppose we have a function $f(x)$ defined on $[a,+\infty[$ and consider the improper integral of first kind $\int_a^{+\infty}{f(x)}{dx}$ where $f$ can be written as $f=f_1+g_1$ so we will have ...
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1answer
13 views

Convergence of the integral of $1/|x|^p$ in $B(0,1)\subset\mathbb{R}^n$

I'm looking for a reference to show the convergence for \begin{equation} \int_{B(0,1)} \frac{1}{|x|^p} \,\mathrm{d}x, \end{equation} where $B(0,1)\subset\mathbb{R}^n$ is the open unit ball, depending ...
0
votes
1answer
30 views

$f:[0,\infty) \to [0,\infty)$ continuous function such that $\int_0^{\infty} f(x) dx$ is convergent ; is $\{f(n)\}$ bounded? [duplicate]

Let $f:[0,\infty) \to [0,\infty)$ be a continuous function such that $\int_0^{\infty} f(x) dx$ is convergent ; is $\{f(n)\}$ bounded ? I know that if $f$ is uniformly continuous then $\{f(n)\}$ ...
0
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2answers
66 views

Showing convergence of the integral $\int_{1}^{\infty}\frac{\ln(x)\cos(x)}{x^2+1}\,{\rm d}x$

I need to show that the following integral either converges or diverges. $$\int_{1}^{\infty}\frac{\ln(x)\cos(x)}{x^2+1}\,{\rm d}x$$ I am fairly certain it converges, but am stuck on showing how. ...
0
votes
0answers
24 views

convergence about the integral of $(\sin x)^a/x^b$

I don't know for which $a>0$ and $b>0$ the following integral exists as a improper Riemann integral. I want to find all such $a$ and $b$. $$\int_1^{\infty} {(\sin x)^a \over x^b}dx$$
3
votes
1answer
44 views

When does this improper Riemann integral converge?

When $n$ is odd, how can we show that $$\int_1^{\infty} \ln\left(1+{{(\sin x)^n}\over{x^c}}\right)dx$$ exists as a Riemann integral if and only if $c \geq { 1 \over 2}$? For the general techs used ...
1
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2answers
57 views

Integral convergence $\int_0^\infty \frac {x^n e^{-x^2}}{1+x^2} $

Show that: $$\int_0^\infty \frac {x^n e^{-x^2}}{1+x^2} $$ Converges for every value of $n$ ($n$ is a natural number). I know how to show that the integrand goes to $0$ as $x$ goes to $\infty$, but ...
0
votes
2answers
48 views

Does the integral converge? $\int_1^\infty \frac{1}{x(x^2+1)}dx$

Does the integral converge? $$\int_1^\infty \frac{1}{x(x^2+1)}dx$$ Well, I did show that it converge by finding the indefinite integral first, and getting to $lim =\frac{ln2}{2}$. Which means it ...
0
votes
1answer
31 views

For which values of $a$ is the integral $\int _2^{\infty }\:\:\frac{\sqrt{x+3}}{\left(x^2-2x\right)^a}dx$ converges? [closed]

I have a problem with this integral, I dont know what method to use to solve it. For which values of $a$ is the integral converges? $$\int _2^{\infty }\:\:\frac{\sqrt{x+3}}{\left(x^2-2x\right)^a}dx$$
2
votes
1answer
61 views

For which $a>0$ does $\int_a^\infty \frac{\mathrm{d}x}{(x^2-a)^{4a}}$ converge?

As the title suggests, I need help finding $a>0$ for which the following improper integral converges: $$\int_a^\infty \frac{\mathrm{d}x}{(x^2-a)^{4a}}$$ So, at first I thought I would just do ...
0
votes
1answer
15 views

Contour integral to real integral: find suitable change of variables

There's probably simple solution but... I have a contour integral of the form $\int _{-i \infty}^{+i \infty} f(t) \ dt$. I want to make a transformation $t = g(s)$ so that the integral is real and of ...
0
votes
1answer
43 views

Value of integral

The value of $$\int_0^\infty t^{-3/2}\left(1-e^{-t}\right)\,dt=$$ Source. Applying integration by parts or using any kind of substitution is not working. My attempt: Splitting the ...
2
votes
2answers
57 views

Integral of Lorentzian type with trigonometric function

Consider the following Riemann integral $$ \int_0^\infty \mathrm{d}x \frac{\alpha^2}{(x-x_0)^2+\alpha^2} \frac{\sin\left[{\left(x - x_1\right) t }\right]}{x-x_1} $$ with the displacements $x_0,x_1 \in ...
1
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2answers
55 views

Comparing the Indefinite Integrals Convergence for $1/x$ and $1/x^{2}$ between 1 and $\infty$.

This is my question. I've been told that $1/x^2$ converges while $1/x$ diverges. My intuition tells me that looking at these just plain out as functions that both should converge...my reasoning is as ...
5
votes
3answers
72 views

Is the integral $\int e^{2 \pi i z^2} dz$ uniformly bounded for any interval of $\mathbb{R}$?

I was wondering if there exists a constant $C$ such that $| \int_I e^{2 \pi i z^2} dz | \leq C $ for any interval $I$ of $\mathbb{R}$? Here I want $C$ to be independent of the choice of the interval ...
-1
votes
1answer
23 views

minimal subnormal extensions of $u$

Q1:Does every $u\in B(H)$ admit a minimal subnormal operator?why? Q2:Two minimal subnormal extensions of $u$ are equivalent.