Tagged Questions
3
votes
1answer
36 views
A delta function conjecture: almost any function can be a delta kernel
I have been thinking about delta kernels, and I think I have come up with a surprising result:
If $f:\Bbb{R}\to\Bbb{R}$ is such that $\int_{-\infty}^\infty f(x)\,dx=L$ is finite and nonzero, then ...
0
votes
2answers
135 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
5
votes
3answers
60 views
Test for convergence for improper integral $1/x^x$
I am having trouble determining if this is convergence or divergence
$$\int^1_0 1/x^x dx$$
4
votes
1answer
60 views
Test of convergence of $\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$
I am having some trouble with this problem and don't know if I am doing it right:
$$\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$$
so the steps I have taken so far are, I split it into
...
1
vote
0answers
29 views
Is $f$ integrable, in the Darboux sense, on $[0,1]$?
Is
$$
f(x)=
\begin{cases}
0,\quad &x=0,\\
x\sin x,&x>0,
\end{cases}
$$
integrable, in the Darboux sense, on $[0,1]$?
I know the Darboux integral has to do with the upper sum and lower sum ...
0
votes
1answer
36 views
is f improperly integrable if g is not
$ f,g $ are nonnegative and locally integrable on $ [a,b) $ and
$ L := \lim_{x\to b-}\frac{f(x)}{g(x)}\ $ exists as extended real number.
If $ 0 < L \le \infty $ and $g$ is not improperly ...
2
votes
2answers
55 views
$\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx$
Prove that: $\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx=\int_{-\infty}^{\infty} -2x e^{-x^2} \sin(2tx) dx$
This is my proof:
$\forall \ t \in \mathbb{R}$ (the improper integral ...
14
votes
1answer
156 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
1
vote
4answers
80 views
check the convergence of the integral $\int_{0}^{\infty}\frac{1}{x\log x}\,dx$
Help me on checking the convergence of the integral $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx$$
I have tried it in this way $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx=\int_{0}^{\frac{1}{2}}\frac{1}{x\log ...
0
votes
2answers
71 views
check the convergence of the improper integral$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$
How to check the convergence of the improper integral$$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$$
I can only check that the integral is divergent for $p\geq1$, help for the cases when $p<1$.
...
0
votes
1answer
50 views
Prove convergence of improper integral using change of variable.
This may be trivial, but I could use some help...
Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
0
votes
1answer
47 views
Problem on Yukawa Potential
One definition of the Yukawa potential on $R^n$ is the solution $G$ in the sense of distributions to $(-\Delta + \mu^2)G = \delta$. This 'green's function' is given by
\begin{align*}
G(x) = ...
1
vote
1answer
91 views
Existence of Riemann-Liouville Integral
The Riemann Liouville integral is defined as:
$\frac{1}{\Gamma\left(\nu\right)}\int\limits _{h}^{t}\left(t-\xi\right)^{\nu-1}f\left(\xi\right)d\xi$
It is supposed it does exist for all $\nu>0$ and ...
2
votes
2answers
27 views
Problems with determining convergence of integral
It should be easy but I'm not sure... For which $\alpha \in \mathbb{R}$ the following integral is convergent:
$$\int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha}dxdy \ \ ?$$
I get for all $\alpha \neq 1,2$ ...
5
votes
3answers
122 views
$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$
If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that
$$ \int^{\infty}_0 |f(x)|^p dx < \infty $$
The integral is with respect to lebesgue measure. Any solution or hints would ...
0
votes
0answers
98 views
Interesting integral related to floor function
Problem.
Evaluate $\displaystyle F(n, k):=\int_{0}^{1} \frac{1-\{1/x\}^n}{1-\{1/x\}^k}dx$ where $n$, $k$ are positive integers. ($\{x\}=x-\lfloor x\rfloor$)
Someone proposed this interesting problem ...
5
votes
2answers
146 views
Prove that $\int_0^{\infty} \frac{x^{2n+1}}{e^{\pi x}-1}\mathrm{d}x \in \mathbb{Q}, \forall \,n\in\mathbb{N}$
Prove that
$$\int_0^{\infty} \frac{x^{2n+1}}{e^{\pi x}-1}\mathrm{d}x \in \mathbb{Q}, \forall \,n\in\mathbb{N}$$
0
votes
0answers
51 views
Improper or Undefined
Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral
$\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined?
If we take it as a legitimate function for improper Riemann ...
2
votes
0answers
83 views
Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$
Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.
$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$
...
6
votes
1answer
128 views
A improper integral with Glaisher-Kinkelin constant
Show that :
$$\int_0^\infty \frac{\text{e}^{-x}}{x^2} \left( \frac{1}{1-\text{e}^{-x}} - \frac{1}{x} - \frac{1}{2} \right)^2 \, \text{d}x = \frac{7}{36}-\ln A+\frac{\zeta \left( 3 \right)}{2\pi ^2}$$
...
3
votes
4answers
86 views
Improper integral depending on three parameters
I am not able to prove that
$$
\int_0^\infty \frac {e^{-\alpha x}-e^{-\beta x}}{x} \sin(\gamma x)\, \mathrm{ d}x = \arctan\left( \frac {\beta}{\gamma} \right) - \arctan\left( \frac {\alpha}{\gamma} ...
3
votes
3answers
169 views
Two improper log integrals
Evaluate
$$\int_0^{\frac{\pi}{2}}\ln ^2(\tan x)\text{d}x$$
$$\int_0^{\frac{\pi}{2}}\ln ^2(\sin x)\text{d}x$$
0
votes
2answers
67 views
Exponential improper integral
How to calculate $$\int_0^\infty \frac{5x}{\theta}\left(1-\text{e}^{\frac{-x}{\theta}}\right)^4\text{e}^{\frac{-x}{\theta}}\text{d}x$$
I have done this one by expanding the terms and integrate by ...
2
votes
1answer
78 views
Is there a closed form or a better solution?
This is another integral in the book "irresistible integral"
I can find that:
$$\int_0^\infty \frac{1}{\left( x^4 +2ax^2+1 \right)^{m+1}} \, \text{d}x =\frac{{{\left( -1 \right)}^m}\sqrt{2}\pi ...
1
vote
1answer
108 views
Application of Frullani integral
Show that :
$$\begin{align}
& \int_{0}^{\pi }{\frac{f\left( \alpha +{{\text{e}}^{xi}} \right)+f\left( \alpha +{{\text{e}}^{-xi}} \right)}{1-2p\cos x+{{p}^{2}}}}\text{d}x=\frac{2\pi ...
0
votes
0answers
60 views
a few questions about the convergence of improper integrals
Let $f(x)$ be a continuous function on $[1,\infty)$ that oscillates between positive and negative values and tends to $0$ as $x$ approaches $\infty$. Will $\int_{1}^{\infty} f(x) \ dx $ always ...
0
votes
0answers
61 views
About Laplace transform
I dont understand the following working, why the integral becomes double integral?
$$\begin{align}
& \ \ \ \int_{0}^{1}{{{\left( \frac{1}{\ln x}+\frac{1}{1-x} ...
30
votes
2answers
607 views
Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$
Prove that
$$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
2
votes
1answer
135 views
An exponential improper integral
Evaluate :
$$\int_{0}^{\infty }{\frac{{{\text{e}}^{-{{x}^{2}}}}}{{{\pi }^{2}}+{{\left( \gamma +x \right)}^{2}}}\text{d}x}$$
1
vote
0answers
168 views
Prove this integral (about gamma function)
Prove that :
$$I\left( a\ ,\ b \right)=\int_{0}^{\infty }{\frac{{{x}^{a-\frac{3}{2}}}}{{{\left( {{x}^{2}}+\left( {{b}^{2}}-2 \right)x+1 \right)}^{a}}}\text{d}x}={{b}^{1-2a}}\frac{\Gamma \left( ...
2
votes
1answer
236 views
Prove that if $f$ is Riemann-Integrable on $ [0,1]$ then $\lim_{c\rightarrow 0} \int_c^1 f(x)dx$ exists and is equal to $\int_0^1 f(x)dx$
The definition of Riemann-integrability states that if $f$ is Riemann integrable on $[0,1]$ then for any $\epsilon$ there exists a partition $P_\epsilon$ of $[0,1] $ such that for any $P\supset ...
21
votes
7answers
673 views
Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$
Evaluate
$$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$$
3
votes
3answers
240 views
how to show that $\int_0^\infty \sin(x^2) dx$ converges [duplicate]
Possible Duplicate:
Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.
What test do I use to show that the following integral converges?
$$ \int_0^\infty \sin (x^2) \; dx$$
1
vote
1answer
92 views
On the convergence of an improper integral
I am interested in finding the values of $a, b$ such that the integral
$$
\int_0^{\infty}\frac{{\left|\log x\right|}^b}{x^a} dx
$$
converges.
My idea was to separate this integral:
$$
...
2
votes
3answers
76 views
asymptotic behavior of function
I'm preparing for my calculus exam. And I need prove this theorem.
Theorem:
Suppose that $f(x)$ is monotonic function at $x\in(a;\infty)$,
integral $\int_{a}^{\infty}f(x)dx$ is converge. ...
1
vote
1answer
89 views
integration by parts/ improper integral question
This is from an old qualifying examination question.
Let $a>1$ be fixed. Show that
$$ \displaystyle A_N=\pi i a \int_1^N t^{a-\frac{3}{2}}e^{\pi i t^a} dt $$
converges to some complex number as ...
6
votes
2answers
114 views
Does $\int_0^\infty \frac{x\arctan x}{\sqrt[3]{1+x^4}}dx$ converge?
I have to determine whether
$$\int_0^\infty \frac{x\arctan x}{\sqrt[3]{1+x^4}}dx$$
converges or not.
I suspect it doesn't because $\arctan x$ is very close to $\pi/2$ as $x$ goes to infinity, and
...
6
votes
4answers
332 views
Evaluate $\int_{0}^{\infty}\frac{\alpha \sin x}{\alpha^2+x^2} \mathrm{dx},\space \alpha>0$
Evaluate
$$\int_{0}^{\infty}\frac{\alpha \sin x}{\alpha^2+x^2} \mathrm{dx},\space \alpha>0$$
I thought of using Feyman way, but it doesn't seem to help that much.
Some hints, suggestions?
Thanks.
1
vote
1answer
127 views
Convergence/divergence of $\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$
Prove that the improper integral $$\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$$ diverge or converge.
4
votes
1answer
136 views
Real Analysis - Conditions for Boundedness of a function
This question appeared on a Mathematics PhD Preliminary Examination - Real Analysis section.
Let $Q=\{{0<x<1, 0<y<1}\}.$ For what values of $a,b$ is the function
$$x^ay^b ...
1
vote
0answers
101 views
comparison test for improper integrals
Let $f$ and $g$ be continuous functions on $(a,b)$ such that $0 \le f\left( x \right) \le g(x)$ for all $x \in \left( {a,b} \right)$; $a$ can be $ - \infty $ and $b$ can be $ + \infty $.
Prove: ...
1
vote
1answer
72 views
Prove $\lim_{n \to \infty} \int_0^1f(x)e^{inx^3}dx = 0$
Assume that $f:[0,1] \to R$ is a smooth function. Prove that
$$\lim_{n \to \infty} \int_0^1f(x)e^{inx^3}dx = 0.$$
Attempt at solution:
I think the solution may require the interchange of limit and ...
0
votes
2answers
72 views
Bounding the integral $ \int_1^\infty \frac{ (\log(y))^n }{y^2} \ dy $
I'm trying to show that the integral
$$
\int_1^\infty \frac{ (\log(y))^n }{y^2} \ dy
$$
is convergent for every real number $ n \geq 1$. If $ n < 2$, I can bound $ |\log(y)|$ by $y$ and hence show ...
2
votes
1answer
170 views
Show that $\int\limits_{-\infty}^{\infty} \! \mathbb{e}^{-x^2} \, \mathrm{d}x$ = $\sqrt{\pi}$. [duplicate]
Possible Duplicate:
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
Show that $\int\limits_{-\infty}^{\infty} \! \mathbb{e}^{-x^2} \, \mathrm{d}x$ = $\sqrt{\pi}$.
...
1
vote
1answer
267 views
improper Riemann integral and Lebesgue integral
Let $f$ be a continuous function on $(0,1]$ and is defined as $f: [0,1] \to \mathbb R$. Show that if $f$ is lebesgue integrable on $[0,1]$, the improper Riemann integral $\lim_{\epsilon \to 0} ...
9
votes
2answers
257 views
$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \mathrm dx$ Evaluate Integral
Evaluate
$$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \ dx$$
2
votes
2answers
430 views
Convergence/absolute convergence of $\int_0^\infty \frac{\cos x}{1+x}dx$ (Baby Rudin P6.9)
Problem 6.9 of Rudin's PMA asks the reader to demonstrate conditions in which indefinite integrals that satisfy the definition
$$\int_a^\infty f(x)dx = \lim_{b\to\infty} \int_a^b f(x)dx$$
can ...
19
votes
2answers
480 views
$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral
Evaluate
$$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$
8
votes
3answers
482 views
Evaluate $\int_{0}^{\infty} \frac{{(1+x)}^{-n}}{\log^2 x+\pi^2} \ dx, \space n\ge1$
Evaluate the integral
$$\int_{0}^{\infty} \frac{{(1+x)}^{-n}}{\log^2 x+\pi^2} \ dx, \space n\ge1$$
4
votes
1answer
135 views
Evaluate the following improper integral with bounds.
I need ideas for solve this improper integral, i know is hard and is a bonus for my analysis course, so i would really appreciate your help, thanks
$$\int_{1}^{\infty}\dfrac{x\sin(2x)}{x^2+3}dx$$
...
