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
36 views

Square integrable harmonic function

Let $u$ be harmonic function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u^2 dx1...dxn < +\infty$ (1). I've got to prove that it implies $u=0$. My idea was to prove that (1) implies that $u$ ...
6
votes
3answers
141 views

Convergence of $\int_{0}^{+\infty} \frac{x}{1+e^x}\,dx$

Does this integral converge to any particular value? $$\int_{0}^{+\infty} \frac{x}{1+e^x}\,dx$$ If the answer is yes, how should I calculate its value? I tried to use convergence tests but I failed ...
0
votes
0answers
37 views

Integrals that don't coincide with the Riemann integral?

This is probably a lame question, but I was wondering if there exist integrals that do not coincide with Riemann's integral for function that are integrable with respect to both these integrals?
0
votes
1answer
40 views

Convergence and Divergence of Improper Integral

Is this integral $\int_{0}^{\infty}x^m (lnx)^n dx$ divergent or convergent? Why? I understand that if we plot the integrand at any values of m,n (except zero), we can clearly see that the limit does ...
2
votes
1answer
25 views

Convergence of a multivariable improper integral

For which values of $a\in\mathbb{R}$ the following integral converges? $\iint_D{\frac{1}{(\sqrt{1-(x^2+y^2)})^{5a}}}dxdy$ $D=\{(x,y):\sqrt{x^2+y^2}<1\}$ My attempt: Changing to ...
0
votes
1answer
36 views

Improper Integrals with an Infinite Limit of Integration

This is from my textbook. the textbook says it makes no sense when $b=\infty$, I think what it want to show us is $\Delta x$ could also be $\infty$ because of $b$, but as $n$ is also $\infty$, it ...
0
votes
1answer
26 views

Convergence of an integral and the asymptotic of the function

Let $f(x):[0,\infty) \to [0,\infty)$. Is it true that $\int_0^\infty f(x)dx<\infty$ implies $xf(x) \to 0,$ as $ x\to \infty?$
0
votes
1answer
28 views

improper integral combination

This is an example from my textbook I'm just thinking why do we have to break the very right-hand side of the integral in red into two parts again, can we make it like $$\lim_{R_{1} \to{1^{+}},R_{2} ...
1
vote
2answers
46 views

improper integrals( very basic)

I don't quite understand what it really means. Do it mean that $f(c_i)$ could be infinite and $dx$ is very small, so you can't determine what the infinite$\times$very small is?
2
votes
2answers
62 views

Improper integral using residue theorem

I am meant to use the residue theorem to show that $\int\limits_{-\infty}^\infty \frac{\cos t}{(t^2+1)^2}dt=\frac{\pi}{e}$. So far I have deduced that I should take a contour over $\alpha$ the path ...
2
votes
1answer
73 views

Evaluation of an integral

How would one prove that: $$\int_0^{\pi/2} \frac{\ln (1+\cos \theta)}{\cos \theta}\, {\rm d}\theta= \frac{\pi^2}{8}$$ This is what I did. \begin{align*} \int_{0}^{\pi/2}\frac{\ln \left ( 1+\cos ...
0
votes
1answer
63 views

Limit of the gamma function

I have to prove that $$\lim_{x\to 0^{+}}\Gamma(x)=\lim_{x\to+\infty}\Gamma(x)=+\infty$$ where $\Gamma$ is the gamma function. As for the $x\to\infty$ I would like to show that $\Gamma$ is increasing ...
0
votes
0answers
44 views

Improper integral of $\int_0^\infty \cos (t^2x) dt$

When I try to do the inverse Fourier transform of $\frac{1}{\sqrt{|k|}}$, an improper integral shows up: $$\int_0^\infty \cos (t^2x) dt = \sqrt{\frac{\pi}{8 |x|}}.$$ Can anyone show why? (Contour ...
2
votes
0answers
21 views

Leibniz rule or not? improper integral? [duplicate]

If $f(x)=(\int_0^x e^{-t^2}dt)^2$ and $g(x)$=$\int_0^1{e^{-x^2(1+t^2)}\over(1+t^2)}dt$ then $f'(\sqrt\pi) + g'(\sqrt\pi)$ is equal to?? i tried to use Leibniz rule and differentiate with respect to x ...
6
votes
4answers
124 views

Evaluate the integral $\int^{\infty}_{0} e^{-x}x^{100}dx$

$$\int^{\infty}_{0} e^{-x}x^{100}dx$$ I am sure is something here I can not see, else it is integration by parts 100 times.
8
votes
1answer
116 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. If $0 < \arg(z) < \pi$ and $|z| \to \infty$ at a ...
1
vote
1answer
27 views

Prove that the condition $x(\tau)>\xi$ of a divergent integral implies that $x(t)>\xi$

Let $ E, J \subset \mathbb R$ be open intervals and let functions $h:J \to \mathbb R$ and $g: E \to \mathbb R$ be continuous. let $\xi \in E$ and assume that $g(\xi)=0$. Define $f:J \times E \to ...
1
vote
1answer
40 views

absolutely integrability implies function approaches zero at positive infinity

Is the following statement true? $$\text{If function $f$ is absolutely integrable on $[0, \infty)$, this implies } \lim_{x \rightarrow \infty} f (x) = 0.$$ If yes then how would I prove it? Note: I ...
0
votes
2answers
39 views

Integral from $-\infty$ to $\infty$ of a function?

Integral from $-\infty$ to $\infty$ of $xe^{-x^2}$. Now I know if the integral from $a$ to $\infty$ of $f(x) dx$ and $-\infty$ to $a$ are convergent, then I could find the integral by summing the ...
1
vote
0answers
45 views

Is there a close form expression for the integral $ \int_a^b |x-c|^n e^{-x^2/2} $

Is there a close form expression for the integral \begin{align} \int_a^b |x-c|^n e^{-x^2/2} dx \end{align} by close form I mean it can be in terms of well know functions such as $Q$-function, ...
-2
votes
2answers
109 views

Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$ [closed]

As a consequence of this Q, I need some help evaluating the following integral: $$\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$$ Integration by parts wouldn't simplify things and I guess that ...
6
votes
1answer
132 views

Improper integral: $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $.

mathematica is reporting that the improper integral $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $ coverges to $2\cos(1)$. However, when I try to confirm this by actually integrating it using ...
2
votes
6answers
128 views

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

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
vote
2answers
61 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
vote
0answers
37 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?
0
votes
0answers
28 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 ...
1
vote
3answers
100 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
vote
2answers
92 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
votes
3answers
96 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 ...
8
votes
2answers
224 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
votes
1answer
75 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
50 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 = ...
0
votes
1answer
56 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
59 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
votes
0answers
36 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
37 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
vote
1answer
23 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
72 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 $W_{T_2}$ ...
4
votes
1answer
65 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
votes
0answers
52 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
vote
1answer
68 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 ...
-2
votes
1answer
58 views

Elementary integral for square roots of trig functions? [closed]

What's an easy way to calculate something like $\int \sqrt{1+\cos x} \text{ d}x$?
0
votes
2answers
60 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} = ...
0
votes
0answers
37 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 ...
0
votes
1answer
36 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: ...
1
vote
2answers
42 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
votes
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
63 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
votes
1answer
51 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
vote
4answers
93 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 ...