2
votes
0answers
45 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
7
votes
0answers
43 views

Real analytic methods for the following integral

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
0
votes
0answers
31 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
1
vote
1answer
38 views

Calculate $\int_0^1f(x)dx$

Calculate $\int_0^1f(x)dx$,where $$\ f(x) = \left\{ \begin{array}{l l} 0 & \quad \text{if $x=0$ }\\ n & \quad \text{if $x\in(\frac{1}{n+1},\frac{1}{n}]$} \end{array} \right.$$ ...
2
votes
0answers
21 views

If $f$ is increasing, then for all $n\in\mathbb{N}$ there exists $P_n$ : $U(f,P)-L(f,P) \leq (b-a)/n$

I've already proven that, if $f:[a,b] \to \mathbb{R}$ is continuous and increasing, with $a,b\in \mathbb{R}$, then $$U(f,P) - L(f,P) = \sum_{i=1}^{n}\left[ f(x_i) - f(x_{i-1})\right](x_i - x_{i-1})$$ ...
3
votes
3answers
81 views

Evaluate integral: $\int_0^{+\infty}\frac{\cos{bx}-\cos{ax}}{x}dx$

It seems that $\displaystyle\int_0^{+\infty}\frac{\cos x}{x}$ is divergent, so how to solve this problem? $$\int_0^\infty \frac{\cos bx -\cos ax}{x}\, dx\quad,\quad\mbox{where}\,a,b>0$$ It's ...
0
votes
1answer
39 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
0
votes
1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
1
vote
3answers
59 views

The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
7
votes
2answers
146 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
0
votes
0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
1
vote
1answer
75 views

A function $f(x)$ that Riemann integrable on $[a,b]$.

Define a function $f(x)$ that Riemann integrable on $[a,b]$. Let $$g(x)=\begin{cases} f(x)&\text{if}&x\in[a,b], \\ f(a)&\text{if}&x<a, \\ f(b)&\text{if}&x>b. ...
2
votes
0answers
40 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
2
votes
1answer
27 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
1
vote
0answers
34 views

Prove $f(x)=x$ is Lebesgue integrable on $[0,1]$

Prove that $f(x)=x$ is Lebesgue integrable on $[0,1]$. My definition of integrable comes from Royden's Real Analysis (4th ed). So $f$ is integrable if the lower integral is equal to the upper ...
0
votes
0answers
34 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
0
votes
1answer
21 views

Darboux integrals with bisected partition

Let us call $\overline{\int_a^b}f(x)dx$ the Darboux upper integral of $f$ and $\underline{\int_a^b}f(x)dx$ the lower one. Let us construct a partition of $[a,b]$ into $2^n$ intervals $[x_{k-1},x_k]$ ...
2
votes
1answer
31 views

Differentiability of the convolution $\int_0^tf(t-s)g(s)\;ds$

Given two continuously differentiable functions $f,g:[0,\infty)\to\mathbb{R}$. I want to know what we can tell about the differentiability of $$(f\ast g)(t)=\int_0^tf(t-s)g(s)\;ds$$ Especially, why ...
5
votes
3answers
123 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
0
votes
3answers
26 views

Semi Gauss integral limit

I am courrently stuck at showing that: $lim_{x \rightarrow \infty}\int_0^xe^{t^2-x^2}dt=0$. Non of my tries by estimations lead to succes so I would appriciate any kind of help.
1
vote
1answer
37 views

Is this map surjective?

Let $B^1(\mathbb{R},\mathbb{R})$ be the set of all locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $$\sup_{t\in \mathbb{R}} \int_t^{t+1}|f(x)|dx<\infty.$$ Consider the map ...
0
votes
1answer
38 views

What can be said about the inverse of the antiderivative of a strictly positive function?

Let $f:\mathbb R\rightarrow [1,\infty)$ be (a strictly positive) function. Define $$F(t) = \int_0^t f(s)ds.$$ Obviously, $F$ is injective and hence invertible. How does $F^{-1}$ look like?
4
votes
3answers
120 views

Integrate $\,\displaystyle\int_0^{\infty } \frac{\cos x}{x} dx$

Although I have known that $\displaystyle\int_0^\infty {{\sin x} \over x} \, dx = {\pi \over 2}$, I have no idea how to work out $\displaystyle\int_0^{ + \infty } {{\cos x} \over x} \, dx$. How can ...
0
votes
0answers
42 views

Guess a property of the integral average value function

Let $f$ be a function that is defined on $[a,b]$ and integrable on $[a,b]$. Def1. $$\hat f(x)=\begin{cases} f(x),&x\in[a,b], \\ f(a),&x<a, \\ f(b),&x>b, \end{cases}$$ ...
0
votes
1answer
18 views

Riemann Stieltjes Integral of discontinuous function

A function $f(x)$ on $[a,b]$ with finite number of discontinuities is Riemann-Stieltjes integrable if $ \alpha $ is continuous where $f$ is discontinuous.In proof we use continuity of $\alpha$ .Can we ...
5
votes
1answer
29 views

Multiple Integration order doesn't agree.

Let $0<x,y,t,z<1$ with the additional condition: $$\begin{align*} x &< t\\ \wedge & \ \\ y &<z \end{align*}$$ Call the set of all $x,y,t,z$ satisfying the above conditions ...
3
votes
2answers
48 views

Prove that $\int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:du=\frac{\Gamma(s)\cos\left(s\arctan \left(\frac{a}{b}\right)\right)}{(a^2+b^2)^{s/2}}$

From the answer of this OP: Ramanujan log-trigonometric integrals, I found the following formula $$\begin{align} & \int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:\mathrm{d}u = \Gamma ...
1
vote
0answers
12 views

Relation between upper sum & Riemann Integration with inequality

Let, $f$ is differentiable function on $[a,b]$ and $f'$ is bounded on $[a,b]$.Then prove that, $|U(P,f)-\int_a^b f(x) dx|<=(b-a)\sup\{|f(x)|:x \in [a,b]\}$, where U(P,f) denotes the upper Riemann ...
13
votes
3answers
246 views

Prove $\int_{0}^{\pi/2} x\csc^2(x)\arctan \left(\alpha \tan x\right)\, dx = \frac{\pi}{2}\left[\ln\frac{(1+\alpha)^{1+\alpha}}{\alpha^\alpha}\right]$

When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by ...
0
votes
1answer
26 views

Quadratic equation form

I have the relation $u=\sqrt{(a_1+b_1t)^2+(a_2+b_2t)^2+(a_3+b_3t)^2} \tag 1$ I need to write $t$ as a function of $u$ ($t=f(u)$). How will I get that ? NB: $a_1,a_2,a_3,b_1,b_2,b_3$ are ...
1
vote
0answers
20 views

Determine whether $\lim_{R\to\infty}\int_0^R\frac{|\sin x|}{x}dx-\frac{2}{\pi}\ln R$ exists

Let $$J(R):=\int_0^R\frac{|\sin x|}{x}dx.$$ (i) Show that $$\lim_{R\to\infty}\frac{J(R)}{\ln R}$$ exists and determine its value (ii)Does $$\lim_{R\to\infty}J(R)-\frac{2}{\pi}\ln R$$ exist? If ...
1
vote
1answer
23 views

$\sum_{n=1}^{[a]} a_n f(n) = - \int_1 ^{a}A(x)f'(x)dx + A(a)f(a) $

Let $\{a_n\}$ be a sequence of real numbers. For $x \geq 0 $, define : $A(x) = \sum_{n=1}^{[x]} a_n$ where $[x]$ refers to the greatest integer function Let $f$ have a continuous derivative in the ...
0
votes
2answers
30 views

$\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$

I am trying to show the following: $\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$ I tried to use Fubini's theorem and the fact that ...
0
votes
0answers
34 views

Monotonic function $f$ - Riemann Stieltjes Integral

If $f \in R(\alpha)$ ( where $\alpha$ is some function ) on $[a,b]$ and if for every monotonic function $f : $ $\int_a ^b f~ d \alpha = 0 $ then, prove that $\alpha$ must be constant on $[0,1]$ ...
1
vote
2answers
61 views

Proof that if $f$ is integrable then also $f^2$ is integrable

Prove this: Let $f :[a,b] \to \mathbb{R}$ be a bounded and integrable function. Show that $f^2$ is integrable too. I'm in trouble with this. Can anyone show how to do it?
0
votes
1answer
28 views

Proving that a solution involving the Laplacian is unique.

I've been asked the following question; If $u$ is a solution of $\nabla^2u = p(x)u$, for $x \in D$, and $\nabla u \cdot n = g(x)$, for $x \in \partial D$, show that $u$ is unique. So, to begin, ...
1
vote
1answer
21 views

Does integrating out a variable in a two-variable measurable function produce a measurable function?

This problem is not a mere consequence of Fubini’s Theorem, so I thought that it would be suitable for posting here on MSE. Let $ (X,\Sigma,\mu) $ and $ (Y,\text{T},\nu) $ denote $ \sigma $-finite ...
0
votes
1answer
27 views

(solved)question about proof 3.8 at the book< Measures, Integrals and Martingales> by Rene Schilling?

I am self studying this book having a following question. At page 18 the last line of proof 3.8 says" since every rectangle I is uniquely determined by its main diagonal" then we reach the ...
0
votes
3answers
25 views

Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann ...
16
votes
3answers
297 views

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

I have spent my holiday on Sunday to crack several integral & series problems and I am having trouble to prove the following integral \begin{equation} \int_0^1 ...
3
votes
2answers
73 views

When $\lim_{n\rightarrow\infty}\int_0^1f(x)\sin(nx)=0$

What is a good sufficient condition for $f:[0,1]\rightarrow\mathbb{R}$ such that: $$\lim_{n\rightarrow\infty}\int_0^1f(x)\sin(nx)dx=0$$ If $f$ is differentiable then by integral by part we can ...
9
votes
3answers
296 views

Evaluate $ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos (2x)}\right) \mathrm{d}x $

Evaluate the following definite integral: $$ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos(2x)}\right) \mathrm{d}x, $$ where $k \in \mathbb{N}_{>0}$.
4
votes
0answers
247 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
0
votes
1answer
18 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
2
votes
1answer
45 views

Proving the equality of a sum and integral.

Taken from Rudin's Real and Complex Analysis text: Suppose $f$ is a continuous function on $\mathbb{R}^1$ with period $1$. Prove that $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n\alpha) ...
1
vote
1answer
55 views

Useful techniques of experimental mathematics (reference request)

I am searching for papers or books that explain thoroughly useful interesting techniques of experimental mathematics that can be understood and profitably applied by an undergraduate student.
3
votes
0answers
41 views

Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
10
votes
1answer
122 views

Evaluating $\int_0^{\pi/2} \int_0^{\pi/2} \frac{\cos(x)}{ \cos(a \cos(x) \cos(y))} dx dy $

Can we avoid the use of the geometric interpretation combined with polar coordinates change of variable for proving that $$\int_0^{\pi/2} \int_0^{\pi/2} \frac{\cos(x)}{ \cos(a \cos(x) \cos(y))} d x ...
4
votes
2answers
70 views

How to compute $\int_0^{\infty} x^{t-1} e^{-x}\ln(x)\,dx$?

I have hit the following integral (in the process of trying to derive a finite-sample correction for the Maximum Likelihood fitting of the Generalized Extreme Value distribution...): ...
6
votes
3answers
153 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...