1
vote
1answer
29 views

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and some condition.

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and ere exists an increasing function $g:[0,1]\rightarrow \mathbb{R}$ such that $$\left|\int_a^b f(x) dx \right|^2 \leq (g(b)-g(a))(b-a)\quad\quad (*)$$ ...
1
vote
1answer
12 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
0
votes
1answer
15 views

$\int_{[0,1]^2}g(y_1-y_2) \Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_{[0,1]}g(m)(1-m)\, dm$

i'm trying to prove the following equality $$\int_{[0,1]^2}g(y_1-y_2) \Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_{[0,1]}g(m)(1-m)\, dm$$ I tried to do the following: ...
0
votes
0answers
55 views

Bound for this integral

Using the fact that $$\sqrt{(1+y^2)} - \sqrt{(1+x^2)} \geq \frac{x}{\sqrt{1+x^2}}(y-x)$$ for each $x,y\in \mathbb{R}$. We need to show that $$L(k)- L(h) \geq \int_a^b \frac{h'}{\sqrt{1+{h'}^2}} ...
0
votes
0answers
16 views

Let $f\in C[0,1]$. Compute $\lim_{t\rightarrow \infty} \frac{1}{t} \log \int_0^1 \cosh(tf(x)) dx$ [duplicate]

Let $f\in C[0,1]$. Compute $\lim_{t\rightarrow \infty} \frac{1}{t} \log \int_0^1 \cosh(tf(x)) dx$. Can anyone give me a hint for this type of problem? I don't know where to start. Thank you!
1
vote
0answers
15 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
1
vote
0answers
57 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
1
vote
1answer
19 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
3
votes
2answers
122 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
6
votes
2answers
175 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
1
vote
1answer
27 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
8
votes
2answers
168 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
1
vote
1answer
30 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
2
votes
1answer
39 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
2
votes
0answers
26 views

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
1
vote
0answers
14 views

Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...
1
vote
1answer
90 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
2
votes
1answer
29 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
1
vote
0answers
14 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
votes
1answer
43 views

If $f$ is in $R[a,b]$, show that

If $f$ is in $R[a,b]$, show that $$\int_a^b f =\lim_{c\to a^+} \int_c^b f$$ the hint is to show $$\left|\int_a^b f - \int_c^b f \right| = \left|\int_a^c f \right| \le \int_a^c |f|$$
1
vote
0answers
16 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
1
vote
0answers
20 views

Proof for Scheffe's Lemma and General Dominated Convergence theorem

While reading this question here about the proof for Scheffe's Lemma, I was confused since someone said the proof in the question was not correct. I thought the argument was fine, and the author only ...
1
vote
2answers
37 views

$\dfrac{\partial}{\partial x}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$

I'm trying to prove the following, interesting, relation: $\dfrac{d}{dx}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$ I tried to integrate by parts the RHS, but i don't ...
7
votes
2answers
186 views

Proving that $\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$

Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible? $$\int_0^1\frac{x \log^2(1-x)}{1+x^2} ...
6
votes
3answers
215 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
5
votes
3answers
106 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
2
votes
0answers
98 views

Definite trigonometric integral

This question is motivated by Iterative Mean, Covariance Algorithm Convergence: Is there a closed form for the integral $$ \int_0^{2 \pi} ...
1
vote
1answer
71 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
2
votes
1answer
35 views

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I'm not sure if I can ...
6
votes
5answers
179 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
9
votes
0answers
149 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
2
votes
2answers
89 views

Do we need the $f,g \geq 0$ condition for $\int f \ d\mu = \int g \ d\mu$?

My lecture notes state the following corollary: Let $f,g \in \mathcal M_\bar{{\mathbb R}}$ (that is, numerical measurable functions), $f=g$ $\mu$-almost everywhere and $f,g \geq 0$. Then $\int f \ ...
11
votes
3answers
339 views

Finding the maximum value of $\displaystyle \int_{0}^{1}e^x\log f(x)dx$ when $\displaystyle \int_{0}^{1}f(x)dx=1$

Suppose that $f(x)\ (0\le x\le 1)$ is continuous and strictly positive and satisfies $$\int_{0}^{1}f(x)dx=1.$$ Then, can we find the maximum and the minimum value of the following? If yes, then how? ...
1
vote
3answers
33 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
4
votes
1answer
32 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
3
votes
1answer
36 views

Anti-derivative is Lipschitz

We know that if a locally integrable function $f:\mathbb{R}\to\mathbb{R}$ is bounded, then the function $F(t)=\int_0^tf(x)dx$ is Lipschitz. In fact ...
2
votes
2answers
159 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
2
votes
2answers
36 views

convergence of a sequence

I'm reading a paper, for proving a claim it defines $$ f_n(x) = \dfrac{(rx-x^2)^n}{n!} $$ when $ r = \frac{a}{b} $ is a rational, and $ I_n = \int^r_0 f_n(x) \cdot \sin x \cdot dx $ , and then it says ...
0
votes
1answer
40 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
1
vote
0answers
52 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
4
votes
2answers
44 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
0
votes
1answer
27 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
5
votes
2answers
155 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
0
votes
1answer
97 views

How does one graph $\sum_{x=0}^{n}$ [closed]

How does one graph a summation, like $$\sum_{x=0}^{n} n$$ Can it be like this Because if you take the points from the summation (0,0), (1,1), (2,3), (3,6) you can tell by summations it only works ...
1
vote
2answers
33 views

Riemann integral of sequence of uniformly bounded functions

Let $f_n : [0,1]\to \mathbb R$ be a sequence of Riemann integrable functions. We call $f$ Riemann integrable if $U(f) = \inf_P U(f,P) = \sup_P L(f,P) = L(f)$ where $U(f,P)$ denotes the upper sum with ...
1
vote
1answer
37 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
2
votes
0answers
52 views

$f\in L^1(0,\infty)$ monotone, show $\lim_{x\rightarrow \infty} xf(x) = 0$ [duplicate]

Here is the solution: First $f$ is monotone and integrable on $(0,\infty)$, wolg we can assume that $f>0$ and approaches $0$ as $x$ goes to infinity. Observe that $$xf(2x) \leq \int_x^{2x} ...
4
votes
1answer
56 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
6
votes
2answers
57 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
6
votes
3answers
137 views

Asymptotic behaviour of the integral of the quadratic mean of the coordinates on the hypercube

I have to compute the limit $\lim_{n\to +\infty}I_n$, where: $$\qquad I_n=\int_{[0,1]^n}\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\,d\mu.$$ I believe that its value is just $\frac{1}{\sqrt{3}}$, since the ...