Questions about the evaluation of specific definite integrals.

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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
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2answers
44 views

An inequality with $a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$

Let the sequence $(a_n)_n$ defined by $$a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$$ 1)Prove that $$\frac12 \leq a_n \leq ...
1
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1answer
24 views

Log Gamma integral

What is the constant $\phi$ in the evaluation \begin{align} \int_{0}^{1/4} \ln\Gamma\left( t + \frac{1}{4}\right) \, dt = \frac{1}{8} \ln(\phi) \end{align} and the constant $\theta$ in the value ...
2
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1answer
30 views

$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}$

Apparently from Mathematica we have: $$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}=\frac{\pi^2\mathrm{sech}^2\left(\frac{\pi t}{2a}\right)}{4a^2}$$ for $a,t$ both real and positive. I ...
2
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0answers
22 views

Closed form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
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37 views

Closed form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

We know a closed-form of the first two powers of the integral of trilogarithm function between $0$ and $1$. From the result here we know that $$I_1=\int_0^1 \operatorname{Li}_3(x)\,dx = ...
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2answers
39 views

Closed form of $\int_0^1 \operatorname{Li}_p(x) \, dx$

While I've studied integrals involving polylogarithm functions I've observed that $$\int_0^1 \operatorname{Li}_p(x) \, dx \stackrel{?}{=} \sum_{k=2}^p(-1)^{p+k}\zeta(k)+(-1)^{p+1},\tag{1}$$ for any ...
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29 views

Calculate a definite integral given the value of another define integral

I'm given that a function $f$ is continuous in $[a, b]$ and given a value $\int_a^b f(u)du = c$. Then I'm asked to calculate $\int_g^h f(t)dt$. I'm looking at the fundamental theorem of calculus but I ...
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1answer
39 views

Examining the convergence of $\int_{1}^{\infty}\frac{1}{x^2+x}\text{ d}x$

I'd like to have my solution verified for this one. I'd like to show that $$\int\limits_{1}^{\infty}\dfrac{1}{x^2+x}\text{ d}x$$ is convergent. Notice, by partial fraction decomposition, that ...
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1answer
18 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
2
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1answer
20 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
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1answer
23 views

Find the area bounded by the given curves

x+y=3 and the coordinate axes. I know how to find the area bounded by 2 curves it's just that I'm confused with "coordinate axes". Is it the same as x=y? or not? please help me understand. okay, ...
3
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1answer
30 views

Reference for closed form integral of $\int_0^1 dz\,z^n/(z-a)$

Is there a closed form (non-recursive) expression for the definite integral $$\int_0^1 dz \frac{z^n}{z-a}, \qquad n\in\mathbb{Z}_+ \text{ and } a\notin (0,1)$$ for general $n$ and $a$ given in terms ...
4
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2answers
65 views

Is this closed form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is ...
4
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2answers
64 views

Proving convergence of $ \int \limits_0^{\infty} \cos\left(x^2\right) dx $

How would one prove the convergence of $$ \int_0^{\infty} \cos\left(x^2\right) \,\mathrm dx $$ I tried using the integral test for convergence by noting that making the substitution $u = x^2$ means ...
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0answers
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find the kernel of voltera 2nd kind with particular form 2. (Alternative approach)

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach) in which we take kernel function of x and t ot just x or just function of t. we try to solve it with alternative ...
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Integral involving logarithm and inverse trigonometric function [on hold]

$${\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)\ln \left( {1 + x} \right){{\ln }^2}x}}{{1 + x}}} },{\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)\ln \left( {1 + x} \right){{\ln }^2}x}}{{1 - ...
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0answers
40 views

help me with this integral $\int_0^{\infty}\frac{1}{x}e^{-kx}\sin xdx=\arctan k^{-1}$ [duplicate]

$\int_0^{\infty}\frac{1}{x}e^{-kx}\sin xdx=\arctan k^{-1}$
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1answer
104 views

Log integrals IV

It can be determined that the integral \begin{align} \int_{0}^{\pi/2} \frac{x}{\sin(x)} \ln\left(\frac{1+\cos(x) - \sin(x)}{1+\cos(x) + \sin(x)} \right) dx \end{align} has a finite value. Is there an ...
5
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1answer
74 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
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1answer
42 views

Integral for $\frac{x}{x^2+1}cosx$

When computing Fourier transformation I came across these integral: $$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$ Can anyone give me some hints on ...
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3answers
24 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.
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0answers
10 views

find resolvent kernel of voltera 2nd kind

Find the resolvent kernel of Voltera integral equation with the following kernels $$k(x,t)=29+6t$$ and $$ k(x,t)=5t-6t^2$$
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2answers
33 views

Substation problem with a simple integral

i have this integral $$ \int {4x\over \sqrt{1+4x^2}} dx $$ and i have tried to solve it by doing like this $$ t=\sqrt{1+4x^2}->t^2=1+4x^2->2tdt=8xdx->tdt=4xdx $$ and im gettin this integral ...
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Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
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2answers
34 views

Taking limit with hyperbolic functions

I have a problem with evaluating $$\sinh^{-1}(C \sinh (ax))\bigg|_{-\infty}^{+\infty}$$ where $C$ and $a$ are real positive constants.
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2answers
109 views

Closed form of $\int_0^1 B_n(x)\psi(x+1)\,dx$

Is there a closed-form of the following integral? $$I_n = \int_0^1 B_n(x)\psi(x+1)\,dx,$$ where $B_n(x)$ are the Bernoulli polynomials and $\psi(x)$ is the digamma function. The motivation of the ...
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0answers
32 views

Evaluating the antiderivative of a particular improper integral

The task is to integrate $$\tau = \int\limits_{-\infty}^{+\infty}\frac{dx}{\sqrt{E - \frac{U}{cosh^2(ax)}}}, E>U$$ but after taking the integral I get $$\tau ...
3
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1answer
110 views

Closed form of $\int_0^{\pi/4} \sin(\sin(x)) \, dx$

Let $I(b)$ is the following integral $$I(b)=\int_0^b \sin(\sin(x)) \, dx.$$ There are some $b$ value for that we know a closed-form of $I(b)$ in term of Struve function $\mathbf{H}_n(x)$. For ...
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2answers
39 views

Reduction formula question

Given that $I_n=\int_0^{\pi/2}x^n\cos x\,dx$, show that $I_n=(\frac {\pi}{2})^n - n(n-1)I_{n-2}$. I wrote $x^n$ as $xx^{n-1}$ and then used the by parts formula twice and then once again on one of ...
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0answers
13 views

Changing a double integral to single integral

I have seen this integration problem in a random process text book. We have the following integral. $\int_{-T}^{T}\int_{-T}^{T}C(t_1-t_2)dt_1dt_2 = \int_{-2T}^{2T}(2T-|\tau|)C(\tau)d\tau$ where ...
19
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1answer
774 views

The Wicked Integral

My brother's friend gave me the following wicked integral with a beautiful result \begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] ...
3
votes
3answers
103 views

Integration: $\int_0^{ + \infty } \frac{\cos x}{x} dx$

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 ...
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0answers
40 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. $$\hat f(x)=\begin{cases} f(x),&x\in[a,b], \\ f(a),&x<a, \\ f(b),&x>b, \end{cases}$$ ...
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0answers
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Definite integral similar to beta function but with exponential negative square root

I'm trying to solve the following definite integral: $\mathcal{I} = \int_0^1dx\ x^{P+k/2-m}(1-x)^me^{-\sqrt{x}}, $ where $P\in\mathcal{N}$ (whole positive numbers and zero), $m\in\mathcal{N}$, ...
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1answer
50 views

integral identity relating to tan(x)

How can I prove the integral identity below? $$ \int_0^{\pi/12}\ln \Big(\cot x \tan^2(3x)\Big)dx=\int_0^{\pi/12}\ln \Big(\tan x\Big(\frac{3-\tan^2x}{1-3\tan^2x}\Big)^2\Big)dx=0 $$ where we ...
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1answer
24 views

Is the given straightforward double integration solution wrong?

A solution in my Student solutions manual proceeds: $$\int_{-3}^3\int_0^\frac{\pi}{2}(y+y^2\cos{x})\mathrm{d}x\mathrm{d}y=\int_{-3}^{3}\left[xy+y^2\sin{x}\right]_{0}^\frac{\pi}{2}dy$$. I thought ...
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1answer
36 views

Calculate the integral of …

$\int_{7}^{10}\sqrt{-40+14x-x^2}dx$ I started off by doing $\int_{7}^{10}\sqrt{(x+7)^2-89}dx$ but I don't know whether that's correct and how I should proceed. Edit: Ok so it should be ...
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4answers
80 views

Prove that $2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$

I need some starting ideas, hints for proving that $$2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$$ I already checked that with Mathematica that numerically says that $$\int_0^1 \ ...
3
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2answers
45 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 ...
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3answers
218 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 ...
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2answers
22 views

Calculus find limit with dominance of one function over another

so I have this math problem, that I can't seem to wrap my head around, I have to find the integral: $$\int_1^{\infty}{\frac{d}{dx}}\left(\frac{3\ln(x)}{x}\right)dx$$ I have no idea where to start... ...
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3answers
29 views

Calculus find out if integral converges or diverges

So I have this math problem, where I am supposed to find out whether or not the integral converges or diverges and solve. $$\int_0^1 \frac{3\,dx}{\sqrt{x}(x+1)}$$ I'm not 100% sure as to figure out ...
0
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1answer
21 views

Calculating the limit of the “$\dfrac{volume}{area}$” ratio for a 2D function

Let's assume that we have a well behaving, continuous function $f(x,y)$ defined on $\mathbb{R^2}$. The double integral $\int_{x_0}^{x_1}\int_{y_0}^{y_1}f(x,y)dxdy$ gives the volume of the space ...
2
votes
0answers
55 views

Integral with cosines and power or upper bound

I need to solve this integral or find its upper bound $$\int_M^\infty \frac{2}{(t^2 \pi^2 + \epsilon^2)^\beta}\sin(\pi x t)\sin(\pi z t)\mathrm{d}t$$ I got to simplify upstairs as $$\int_M^\infty ...
0
votes
1answer
31 views

Find the resolvent kernel of particular form of volterra integral equation of second kind

Find the resolvent kernel for the integral equation $$u(x)= 30 +6x+∫_0^x(5t-6t^2 )u(t)dt$$ I try to solve but its lengthy $$k(x,t)=(5t-6t^2)$$ $$k_2(x,t)=∫_ξ^x (5t-6t^2)(5ξ-6ξ^2)dt$$ its very ...
0
votes
0answers
32 views

Properties of a real function defined implicitly by an equation with $\int_0^x e^{f(t)}\mathrm{d}t+i x\int_0^1 e^{f(x-xt)}\mathrm{d}t$

Let $f : \mathbb{R_{\geq 0}} \rightarrow \mathbb{R_{\geq 0}}$ and $\forall x$ let a complex $z$ such that; $$z=\int_0^x e^{f(t)}\mathrm{d}t+i x\int_0^1 e^{f(x-xt)}\mathrm{d}t$$ and ...
1
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1answer
37 views

Solution of the integral $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{3xy}{(x^{2}+y^{2}+z^{2})^{5/2})}e^{i(k_{x}x+k_{y}y)} dx dy$

I'm trying to solve the following integral: $$\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}\frac{3xy}{(x^{2}+y^{2}+z^{2})^{5/2})}e^{i(k_{x}x+k_{y}y)} dx dy$$ i'm using the solution ...
1
vote
3answers
38 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
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1answer
50 views

Log integrals III

The integral \begin{align} J_{m} = \int_{0}^{1} \frac{t^{m}}{1+t} \, \ln(1+t) \, dt \end{align} has the general form \begin{align} J_{m} = (-1)^{m} \left[ A_{m} - B_{m} \, \ln(2) + C_{m} \, ...