3
votes
0answers
28 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
5
votes
1answer
62 views

Problem with a sequence with multiple integrals [duplicate]

How to compute the following limit, $\displaystyle \lim\limits_{n \to \infty} \int_0^1 \int_0^1 \ldots \int_0^1 \sin \bigg(\frac{x_1+x_2+\ldots+x_n}{n}\bigg)\,dx_1 \,dx_2 \ldots \,dx_n$ ? I will ...
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
1
vote
1answer
12 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
1
vote
1answer
39 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
0
votes
0answers
6 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
7
votes
3answers
670 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
2
votes
1answer
34 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
0
votes
0answers
57 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
1
vote
0answers
35 views

Higher-dimension integrability (over rectangles) well-defined

Here is the problem and my work toward a proof: Question: Prove that in the following definition, the value of $\int_E f dx$ is independent of the choice of rectangle $J$: Definition: ...
0
votes
0answers
16 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
1
vote
0answers
17 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
2
votes
1answer
72 views

If $f \in C^\infty$, and $f$ is nonnegative and integrable in $\mathbb{R}$, can I say that $f^\prime$ is integrable?

I'm not sure how to describe the question any further in the title than it is, but I will try to explain what I have done. If $f$ is a Schwartz function, I believe that $f^\prime$ will always be ...
0
votes
1answer
31 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
2
votes
1answer
39 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
1
vote
1answer
21 views

Double integral And polar coordinate system

I have to evaluate this integral over the domain D The Plot would be like this: I decided to use polar coordinate system using it It gives me this but I don't know the upper limit of ...
1
vote
2answers
38 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
0
votes
1answer
26 views

Сhange the order of integration in the double integral

I have to change the order of integration in this double integral I've decided to divide it in two similar areas D1 and D2 And I've got the following result Can You chech it and state my ...
4
votes
2answers
92 views

What are integrating factors, really?

I can follow the rationale for integrating factors well enough, but they still feel like voodoo to me. Every single description of integrating factors I've seen (and I've seen quite a few, including ...
6
votes
1answer
65 views

On derivatives that are not Riemann integrable

Let $f:\;[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$. It is not a mystery that $f'$ need not be Riemann integrable. In fact even if we require $f'$ to be bounded the implication is still false. ...
3
votes
1answer
55 views

Integral inequality problem

Let $f:[0,1]\to\mathbb R$ be a differentiable function with $f(0)=0$ and $f'(x)\in(0,1)$ for every $x\in(0,1).$ Show that $$\left(\int_0^1f(x)dx\right)^2>\int_0^1(f(x))^3dx$$ I am not even sure ...
1
vote
1answer
78 views

Gaussian Quadrature -Deriving a Formula-

eThe following is an exercise in the problem section of the Gaussian Quadrature chapter. The theorem: Derive a formula of the form $$\int_{a}^{b} f(x)dx \approx w_0f(x_0) + w_1f(x_1) + w_2f'(x_2) ...
0
votes
0answers
23 views

proof of an relation

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ et soit $v \in H^1_0(\Omega)$ and let $h \neq 0$. Let $$D_h v = \dfrac{v(x+h,y) - v(x,y)}{h}$$ The questions are: 1- Prouve that ...
0
votes
1answer
36 views

Riemann-integrable functions and pointwise convergence

Hello, I was hoping for some advice on finding a function which will satisfy this. I think I am okay with the actual execution of the answer, but I don't know how I'm supposed to find a suitable ...
1
vote
2answers
72 views

a question about improper integral, I cannot solve it!

If $f(x)$ is continuous in $[0,\infty)$, and $\displaystyle\int_c^{\infty}\frac{f(x)}{x}\, dx$ is convergent for any $c>0$. Please prove $\displaystyle\int_0^{\infty}\frac{f(\alpha x)-f(\beta ...
1
vote
2answers
43 views

Can somebody help me solve the proof about improper integrals?

If $f(x) > 0$ is continuous at $[0, +\infty]$ and $\displaystyle \int_0^{+\infty} \frac{1}{f(x)} dx$ is convergent, please prove $\displaystyle \lim_{\lambda \to \infty} \frac{1}{\lambda} ...
1
vote
0answers
45 views

Integration question measure theory

For the function $$ f(x) = \begin{cases} \infty & \text{if $x=0$} \\ 1/x & \text{if $x \in \mathbb{Q} \smallsetminus 0$} \\ 0 & \text{Otherwise} ...
0
votes
1answer
19 views

question about line integral in a strength field

Let the field strength $\bar{F}(x,y) = e^xy ·\vec{i} , ((x,y) \in R^2)$. How can I prove, without doing any calculations, that the line integer of $\bar{F}$ along the segment joining $(2,0)$ with ...
0
votes
1answer
37 views

How to get a parametrization for a line?

I am currently studying for some exams about line integrals. And I am struggling to get the parametrization for given problems. For example the line of the following Ellipse $ {x^2 \over a^2} + {y^2 ...
1
vote
0answers
27 views

Integration by parts on a sum of a product of partial derivatives

Let $\Omega$ be a region $\subset \subset \mathbb{R}^{n}$, and suppose that both $u_{\infty}$ and $\varphi$ $\in C_{0}^{\infty}(\Omega)$. Then, I want to perform integration by parts on the ...
1
vote
2answers
47 views

Can an integral be proved to have a finite value if an upper bound of the integrand has a finite value for improper integrals?

Can we say $ \int_{0}^{\infty} f(x) \text{dx} < \infty$ if $\exists \quad g(x) : \quad g(x)\geq f(x)\; \forall x \in \mathbb{R}$ and $ \int_{0}^{\infty} g(x) \text{dx}$ is finite. If yes, ...
1
vote
0answers
16 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
28 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
1
vote
1answer
40 views

Signed measure defined by an integral

Let $ (X,\mathcal{M},\mu)$ be a measure space and let $ f:X\to[-\infty,+\infty]$ be an integrable function (i.e. at least one of $ f_+ $ and $ f_-$ is integrable). I want to prove that $ ...
2
votes
2answers
73 views

Does $dx$ in the formula $\int f(x)dx$ represents a differential of x?

While I asked a question about integrals Is $dxdy$ really a multiplication of $dx$ and $dy$?, I found out that many of the answers were assuming that dx is just a notation in the formula $\int ...
0
votes
1answer
35 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
5
votes
0answers
125 views

Showing some complicated integral expression is bounded

In my research, I come across some expression I need to bound. I wish to show that the following integral is bounded in $t$ and $x$, i.e. the following supremum is finite: $$\sup_{t,x\in ...
1
vote
0answers
71 views

Question about integration [duplicate]

I have this, and I don't understand how to do the change of variable. Please help me Thank you.
13
votes
1answer
203 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
8
votes
1answer
152 views

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
0
votes
1answer
30 views

Din derivatives and fundamental theorem of calculus

I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function ...
2
votes
0answers
117 views

integral $I=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
0
votes
2answers
49 views

Integrals on unlimited sets

How do you evaluate this expression $$ \left| \int_{1}^{\infty} 1 \; dx - \int_{1}^{\infty} 1 \; dx \right| \quad ? $$ Using improper integral definition, this should be an indeterminate $\infty - ...
0
votes
0answers
35 views

Derivative and integration

I have an application $f$ such that $f\in C^1([0,1]\times \mathbb{R},\mathbb{R})$ and i have this estimation $$m^4\pi^4+\varepsilon_0\leq f'_u(t,0)\leq (m+1)^4\pi^4-\varepsilon_0~\text{for all}~ t\in ...
5
votes
1answer
70 views

Why does the same limit work in one case but fail in another?

The following questions has been bugging me since high-school calculus. Please help me find my peace once and for all: Consider a revolution solid generated by rotating a nice curve $f(x)$ around the ...
1
vote
1answer
26 views

Geometric question involving integral of a function and its inverse.

I am given a function $\phi(s)$, continuous and strictly increasing with $\phi(0) = 0$, and want to show that for all $a,b \geq 0$, $$ab \leq \int_0^a \phi(x)dx + \int_0^b \phi^{-1}(x)dx.$$ I know how ...
0
votes
1answer
25 views

Integral Estimate Using a Function and its Inverse

I want to show the following: given a measure space $(X,\mu)$ and $f,g$ $\mu$-measurable functions on $X$, $$\int_X |f(x)g(x)| d\mu(x) \leq \frac{1}{2}\int_{|f(x)| \leq 1} |f(x)|^2 d\mu(x) + ...
-1
votes
1answer
175 views

Why?$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1] $

why: $$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1] $$ how to get this ? Please help me Thank you.
1
vote
0answers
32 views

estimate for the integral of the complex exponential of a polynomial

Let $F(t)$ be a polynomial of degree $>0$ and $R>0$ a real number; Is it true that for some constant $c$ independent of $R$, $$\left| \int_{-R}^{R}\exp(2i \pi F(t)) dt\right| < c ...
0
votes
1answer
16 views

Determining function given it's domain, that it is uniform, and it evaluates to as a double integral?

I have a uniform probability distribution with density function $f(x,y)$ such that $$\int_0^2 \int_0^2 f(x,y)dy dx = 1$$ Now I know that $f(x,y)=\frac{1}{4}$ simply by considering the dimensions of ...