0
votes
0answers
19 views

integration formula

help me please all these functions are regular. How we can found this formulation $$ \displaystyle\int_{\Omega} (f(u)-f(k)) \nabla p(g(u)-g(k)) \xi dx = - \displaystyle\int_{\Omega} H(u,k) \nabla \xi ...
3
votes
4answers
90 views

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24} \end{equation} I tried to use by parts method and ended with \begin{equation} \int \ln^2(\cos ...
5
votes
0answers
53 views

How prove $f(a_{i})=0$ if $\int_{0}^{1}x^kf(x)dx=0,k=1,2,3,\cdots,n$ [duplicate]

let $f(x)$ is continuous on $[a,b]$,and such $$\int_{0}^{1}x^kf(x)dx=0,k=0,1,2,3,\cdots,n$$ show that: there exsit $n+1$ different $a_{1},a_{2},\cdots,a_{n},a_{n+1}(a_{i}\neq a_{j},\forall ...
5
votes
1answer
52 views

Convergence of Integral near 0

I am trying to determine the convergence of the integral \begin{equation} \int_0^1 \frac{f(x)}{x}\, dx \end{equation} given that $f(x)$ is bounded and continuous on $[0,1]$, and that $f(x)=0$. The ...
0
votes
3answers
117 views

How to evaluate $\int{d(y^2)}$?

Can anybody help me to solve this integral please: $$\int{dy^2}$$ Here $dy^2$ means $d(y^2)$, not $(dy)^2$. Thanks for any help.
6
votes
1answer
96 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
2
votes
1answer
30 views
+50

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
1
vote
0answers
24 views

Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
0
votes
2answers
59 views

A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
8
votes
2answers
178 views

Integral inequality: $\def\intd{\,\mathrm d}\int_a^b(f'(x))^2\intd x-2\big(f(a)+f(b)\big)^2\geq\frac8{(b-a)^2}\int_a^b(f(x))^2\intd x$

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) \intd x = f\left(\frac{a+b}{2}\right) = 0$$ ...
1
vote
1answer
73 views

Prove that $\int_{-1}^1P_n^2(x)dx=\frac{2}{2n+1}$, where $P_n(x)$ is a Legendre polynomial.

Using Rodrigues' formula and integrating by parts $n$ times, prove that $$\int_{-1}^1P_n^2(x)dx=\frac{2}{2n+1}$$ where $P_n(x)$ is a Legendre polynomial. I tried this way Let $$f(x)=(x^2-1)^s$$ ...
10
votes
4answers
144 views

Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute \begin{equation} \lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx \end{equation} According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
1
vote
1answer
30 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
0
votes
0answers
55 views

Equality case in Hölder's inequality

How can I show that $$\left(\int{p(x)^{1-\sigma}\mathrm dx}\right)^{\frac{1}{1-\sigma}}\cdot \left(\int y(x)^\frac{\sigma-1}{\sigma}\mathrm dx\right)^{\frac{\sigma}{\sigma-1}}=\int p(x) ...
1
vote
4answers
69 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
5
votes
1answer
105 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
1
vote
1answer
25 views

Integrating a Dirac delta function with the argument dependent of a parameter

How can I handle the integral $$ \int_{t_1}^{t_2} \delta(D - x(t)) dt, $$ with $D$ a constant. I want to do a change of variables to perform the integral over $x$ but I am not sure how to proceed.
1
vote
3answers
50 views

Negative functions and their anti-derivatives

Let $f(x)$ be a continuous function with antiderivative $$F(x)=\intop f(x) \mathrm{d}x,$$ where the constant of integration is zero. Does the following hold true: ...
1
vote
0answers
34 views

Determining the sets of alpha for which some (Riemann, Lebesgue - integrals) exists

$$\int_0^{\infty} \frac{\sin(x)}{x^{\alpha}} \, dx.$$ $$\int_{[0, \infty]} \frac{\sin(x)}{x^{\alpha}} \, d \lambda(x).$$ $$\int_{\Bbb R^2} \frac{\sin(\| x \|)}{\| x \|^{\alpha}} \, d \lambda_2 ...
2
votes
3answers
201 views

Why are variables in integration by substitution so counter intuitive?

Integration by substitution is defined as something like $\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$ But for my taste, the variables $x$ and $t$ are exactly ...
2
votes
0answers
34 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
6
votes
1answer
132 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
2
votes
2answers
55 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
0
votes
2answers
103 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
16
votes
2answers
449 views

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
2
votes
3answers
82 views

Calculate $\int\frac{dx}{x\sqrt{x^2-2}}$.

The exercise is: Calculate:$$\int\frac{dx}{x\sqrt{x^2-2}}$$ My first approach was: Let $z:=\sqrt{x^2-2}$ then $dx = dz \frac{\sqrt{x^2-2}}{x}$ and $x^2=z^2+2$ $$\int\frac{dx}{x\sqrt{x^2-2}} ...
2
votes
1answer
55 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
1
vote
1answer
53 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
4
votes
0answers
88 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
0
votes
1answer
18 views

Change of variables theorem in the case $L^1_{\textrm{loc}}(U)$?

I'm trying to write a version of the change of variable theorem for the case of locally integrable functions on open subsets of $\mathbb R^n$. Statement: Let $U, V\subseteq \mathbb R^n$ be open sets ...
0
votes
1answer
42 views

An issue with $\infty \cdot 0$ in showing that Cartesian product of a set with a null set has measure zero

Here is the problem: Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be $\sigma$-finite measure spaces. Furthermore $A\in \mathcal A$ and $N\in \mathcal B$ such that $\nu(N)=0$. Let ...
1
vote
2answers
107 views

Integral Involving Dilogarithms

I came across the identity $$\int^x_0\frac{\ln(p+qt)}{r+st}{\rm ...
12
votes
5answers
299 views

The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$

Prove that $$\int_0^1\frac{\ln(1-x^2)}{x}dx=-\frac{\pi^2}{12}$$ without using series expansion. An easy way to calculate the above integral is using series expansion. Here is an example ...
1
vote
0answers
31 views

Is the “difference-integral” of an integrable function integrable?

assume $f$ is Lebesgue integrable (on $\mathbb{R}$) and $h >0$. Is it true that then the double integral $\int_{-\infty}^{\infty} \int_{x}^{x+h} f(u) \, du \, dx$ always exists? Intuitively I have ...
9
votes
3answers
128 views

Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

I found a nice formula of the following integral here $$\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$$ It states there that ...
2
votes
2answers
90 views

How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$?

is there a way to explicitely evaluate this integral for natural numbers $n,m$: $$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$ Apparently, if $n$ is odd, this integral is zero due to symmetry.
9
votes
5answers
280 views

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

This integral comes from a well-known site (I am sorry, the site is classified due to regarding the OP.) $$\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx$$ I can calculate the integral using the help of ...
11
votes
3answers
206 views

Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

Prove that (please) $$\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$$ I've tried using Taylor series and I ended up with $$-\sum_{m=0}^\infty\sum_{n=1}^\infty\frac{2}{n(m+n+1)^3}$$ I am ...
0
votes
1answer
30 views

Riemann integral of a certain function

Let $\{q_n\}$ be an enumeration of the rationals in $[0,1]$. Define $f$ on $[0,1]$ by $f(x):=\sum_{q_n<x}2^{-n}$. It has already been shown by Lebesgue's Criterion that $f$ is Riemann integrable. ...
1
vote
1answer
46 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
2
votes
0answers
94 views

Line integral $\int_{\gamma}e^xcos(y)dx+e^xsin(y)dy$

First of all, I'm having trouble evaluating $$\int_{\gamma}e^xcos(y)dx+e^xsin(y)dy$$ where $\gamma$ is the triangle with vertices $(0,0), (1,0), (1,\frac\pi2)$ This is what I've done so far: 1) With ...
6
votes
4answers
173 views

How to find $\int_{0}^{\pi/2} \log ({1+\cos x}) dx$ using real-variable methods?

How do you find the value of this integral, using real methods? $$I=\displaystyle\int_{0}^{\pi/2} \log ({1+\cos x}) dx$$ The answer is $2C-\dfrac{\pi}{2}\log {2}$ where $C$ is Catalan's constant.
0
votes
1answer
68 views

Negative part of the integrand in an iterated integral

Hi everyone: Suppose that $(X,\mathfrak{M},\mu)$ and $(Y,\mathfrak{N},\nu)$ are two measure spaces and $f(x,y)$ is an extended real valued measurable function on $X\times Y$. Suppose we can not apply ...
0
votes
0answers
27 views

Uniformly continuous function which is integrable but does not have a limit [duplicate]

Is there an example of a function $f:[0,+\infty)\to \mathbb{R}$ which is uniformly continuous and $\int_0^{+\infty}|f(x)|dx<+\infty$, but $\lim_{x\to+\infty}f(x)\neq0$ (since it is integrable this ...
1
vote
1answer
38 views

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

I found that the following converges where ${0<p<1}$,and ${0<q}$, but I'm having some trouble where q is negative. Because it has some "blow up" point, it seems to diverge, but i'm not ...
1
vote
2answers
31 views

Integral related to the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$

I have difficulty with the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$ How do we go from left to right? Thanks!
2
votes
0answers
53 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
3
votes
0answers
56 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
3
votes
1answer
32 views

an integral estimate from Stein's book, Singular Integral

I am reading the Stein's book Singular Integrals and Differentiability Properties of Functions. In the text (page 40), he states that $$ \int_{|x|\geq 2|y|}\Big|\frac{1}{|x-y|^n}-\frac{1}{|x|^n}\Big| ...
0
votes
1answer
36 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...