All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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168 views

$\int \frac{\tan x}{x} dx$

Evaluate the integral: $$\int \frac{\tan x}{x} dx$$ I tried integration by parts, got stuck. Ideas/ suggestions please.
3
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1answer
72 views

When does $\lim\limits_{n\to\infty}\int_{b}^{a_n}f_n(x)dx=\lim\limits_{n\to\infty}\int_b^\infty f_n(x)dx$ hold?

Let $\{a_n\}\subset \mathbb{R}$ be sequence and $$f_n:[b,\infty)\longrightarrow \mathbb{R}, \qquad n=1,2,\dots .$$ Assume that $$\lim_{n\longrightarrow\infty}a_n=+\infty.$$ Obviously, from the ...
3
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2answers
540 views

From Faraday's law in integral form to the differential form(=Maxwell equation) by using Stokes

I want to understand how Stoke's theorem shows that the integral form of Faraday's law: $$\int_{c(A)} E dr = -\frac{1}{c} \frac{d}{dt} \int_A B ds$$ ($A$ is a surface and $c(A)$ its boundary curve) ...
3
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1answer
114 views

Dense subspaces

How does one go about proving the following statements? (a) $\operatorname{Lip}[a,b]$ functions are dense in absolutely continuous functions on $[a,b]$ under the variation norm - (Another doubt: what ...
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1answer
127 views

Limit and integral

Let $\Omega = \mathopen]0,1\mathclose[$ and let a function $A_n: \Omega \to \mathbb R$ defined as: $$A_n(x) = \begin{cases}\alpha &\text{if } k \epsilon \leq x < (k+\tfrac{1}{2}) \epsilon \\ ...
3
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1answer
114 views

Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$

Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
3
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1answer
77 views

Evaluation a this integral

If $f$ ia a continuously differentiable function on the unit circle and $$ g(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{f(x+t)-f(x-t)}{2\tan\frac{1}{2}t}dt $$ evaluate $$ ...
3
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1answer
212 views

Riemann Stieltjes integral definition and implications

I am studying the Riemann Stieltjes on Tom Apostol's book mathematical analysis second edition and I have a the following question. Given $[a,b]$ we define a partition of this interval to be a set $P ...
3
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1answer
104 views

Definition of an Integral over a Torus

By $\mathbb{T}^1$ I mean the one-dimensional Torus which is the same as the circle $S^1$. How does one make sense of: $$\int_{\mathbb{T}^1}f dm = ?$$ (1) What is the definition of $\mathbb{T}^1$ ...
3
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1answer
81 views

Convergence Theory

Using convergence theory, show that $$\int_{0}^\infty \left(\frac x{x^3 +1} \right) dx$$ is convergent. I think it might be a typo since taking the integral to zero shows divergence in most ...
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2answers
163 views

Limit of an integral with a periodic function

Let $f , g$ be continuous functions: $f:[0 , 2\pi]\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$. Assume $\forall x\in\mathbb{R}:g(x+2\pi)=g(x)$ and $$\int\limits_{0}^{2\pi} \! {g(x)} ...
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1answer
77 views

Fundamental theorem of calculus 1 where integrand is a 2nd order partial derivative

I have a function $b(x,y)$ such that $b(x,0)=0$. Now, suppose I wish to evaluate the following integral: (Note that $b$ is continuous almost everywhere but it is assumed that it is integrable. Also, ...
3
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1answer
92 views

$\int \frac{\sqrt x}{1+\sqrt[4]{x-b}}dx=?$

One of my friends gave me to solve the following integration problem: $\int \frac{\sqrt x}{1+\sqrt[4]{x-b}} dx=?$ where $b$ is any arbitrary constant. I tried to solve it by putting ...
3
votes
2answers
97 views

$\int fg = \lim_{n \rightarrow \infty} \int f_n g.$

From an old notebook of mine, I saw this unproved exercise in class: Let $f_n$ be a uniformly bounded sequence in $L^p$ ($1<p<\infty$) such that $f_n \rightarrow f$ a.e. in $L^p$. Then for ...
3
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1answer
308 views

Differentiation under the double integral sign

Working on three-body dispersion forces I got the following quantity: $$\frac{\partial } {{\partial \lambda }}\int\limits_\lambda ^{\pi - \lambda } {d\theta } \int\limits_\lambda ^{\pi - \lambda } ...
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1answer
40 views

Integral and measurability of a function

Consider the function $f:[0,1] \rightarrow [0,\infty)$ that is defined as follows: $$f(x) = 0 \text{ if $x$ is rational and } 2^n \text{ when $x$ is irrational}$$ Here $n$ is the number of leading ...
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1answer
556 views

How does differentiation under the integral sign work?

From what I gather, it looks like you can use the method when your function depends on a variable and also a parameter. If you are given some definite integral that depended only on a variable, how ...
3
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1answer
67 views

Need help calculating $\int_{0}^{\infty} \frac{1- \cos (t)}{t^\alpha} {\rm d}t$

As part of an larger assignment I need to calculate the following integral $$ \int_{-\infty}^{\infty} \frac{1-\cos(\lambda x)}{|\lambda|^\alpha} {\rm d}\lambda \quad x \in \mathbb{R}, \,1 < \alpha ...
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1answer
123 views

multiplying $\frac{dy}{dx} + \frac{2x+ x^2 + y^2}{2y}=0$ by integrating factor $2ye^x$

I've been introduced to a new concept called the integrating factor and I unsure how it effects the separation of variables for this equation. Can anyone help... By multiplying the following ...
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1answer
54 views

Find a function that that makes the value of this improper integral equal to 1.

I have the following integral: $$I(t) = \int_{0}^{t} \sqrt{1- \frac{a(x)^2}{c^2}}dx$$ where $a(x)$ is some continuous function of $x$, and $c$ is a constant. Also $a(x) < c$ for all $x >0$. ...
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1answer
155 views

Integrals of Hermite polynomials over $(-\infty, 0)$

Does there exist a simple expression for integrals of the form, $I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$, where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th ...
3
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1answer
152 views

Evaluation of integral involving $ \tanh(ax) $

Is it possible to evaluate in closed form the integral $$ \int_{-\sqrt{x}}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr=2\int_{0}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr$$ here $a$ is a ...
3
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1answer
198 views

Integration Antiderivative vertical bar [duplicate]

Possible Duplicate: What is the name of the vertical bar? When taking a definite integral, the first step is finding the anti-derivative. Once you have gone through all the steps to ...
3
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1answer
263 views

Calculating a difficult integral

For $p>0$, let $g(x)=\begin{cases} p\left[\dfrac{x}{p}\right]+\dfrac{p}{2}& x\ge 0,\\ -g(-x)& x<0 \end{cases}$ Try to prove that for all $x \in \mathbb{R}$ ...
3
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2answers
708 views

Proof that monotone functions are integrable with the classical definition of the Riemann Integral

Let $f:[a,b]\to \mathbb{R}$ be a monotone function (say stictly increasing). Then, do for every $\epsilon>0$ exist two step functions $h,g$ so that $g\le f\le h$ and $0\le h-g\le \epsilon$? Does ...
3
votes
1answer
214 views

Differentiation with respect to integral boundary

Using the chain rule show the following proposition: Let $f$ be continuously on $[a,b]$ and $g:J\to[a,b]$ continuously differentiable for an interval $J$. We write ...
3
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2answers
230 views

The volume of the solid from the region bounded by $x=9-y^2$, $y=x-7$, $x=0$ about $y=3$ using cylindrical shells.

The volume of the solid from the region bounded by $x=9-y^2$, $y=x-7$, $x=0$ about $y=3$ using cylindrical shells. I've tried creating two separate regions: $V_1=2\pi(3-y)(9-y^2)dy$ from 3 to 1 ...
3
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1answer
154 views

Integration Techniques

We just learnt the different types of integration techniques in school such as substitution, by parts, etc. But, these methods seem kind of laborious. Do professional mathematics and theoretical ...
3
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1answer
489 views

Partial Derivative of an Integral

If $f(t)$ is a deterministic function of $t$ and $B_{n}$ is a brownian motion and: $Z =\int^t_0 f(s)dB(s)$ How does one take the partial derivatives wrt to $t$ and $B_n$ on an integral like this? I ...
3
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3answers
99 views

understanding change of variable

the following is drawn from a rather rough set of lecture notes and I am not sure I understand it. the goal is to determine for which values of $p$ we have $$ \int_{|x|\leq 1} \frac{1}{|x|^p} \,dx ...
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2answers
202 views

how can I evaluate this integral?

how to evaluate this integral: $$l(y)=\int\limits_\beta^\infty \theta\exp(-y\theta)\alpha\exp(-\alpha\theta) \, d\theta$$ where $\alpha,\beta,\theta,y>0.$ Because I find it infinity! Can anyone ...
3
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1answer
77 views

Integration questions on $\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$ and $\int \frac{x^4}{x^4+5x^2+4} \, dx$

I would appreciate any hints on how to solve the following integration problems, they are my homework questions btw: $$\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$$ $$\int ...
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votes
1answer
84 views

Uniform integrablity of measurable functions

How can I show that if family of $f$ is uniformly integrable then so is {$|f|$}? $($by uniformly integrablity: $\forall \epsilon>0 \ \exists \delta>0: |\int_Ef|<\epsilon,\mu(E)<\delta)$ ...
3
votes
1answer
107 views

Negative integral on intervals implies negative function?

Let $f \in L^1([0,1])$ be such that for all $t \geq s$, $\displaystyle \int_s^t f(u)du \leq 0$. Is it true that $f\leq 0$ almost everywhere?
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votes
1answer
77 views

Can it possible to calculate this integral directly, probably concerning ellipti integral

A student asked me to help her calculating this problem: Assume the length $L$ of a curve is given, and the equation of the curve is \begin{gather*} y(x)=A \sin\Big(\pi x-\frac{\pi}{2}\Big), 0\leq ...
3
votes
1answer
196 views

Integral of a curve with respect to its curvature?

I've been struggling with this one for about $3$ weeks: What is the integral of a $\mathbb{R}^3$ curve with respect to its curvature? I though about approaching it with the Ferret-S formulas, and ...
3
votes
2answers
230 views

How to evaluate $\int \frac{\sin (\pi x)}{|x|^a + 1} dx$

How do I integrate this? $$\int \frac{\sin (\pi x)}{|x|^a+1} \, dx$$ I really struggle to find a solution. I even tried Wolfram Alpha and Mathematica, but neither could give me an answer. I have ...
3
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1answer
39 views

Existence of sequence for an integral function in $\mathbf{R}$.

Is it true that for a function $f\in L^1({\mathbf R})$, there exists a sequence $\,x_n\rightarrow\infty$ such that $x_nf(x_n)\rightarrow 0$?
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1answer
207 views

Can the sign of a continuous function be 'made Riemann integrable'?

Question: I would like to know whether the following statement is true: For every continuous function $f:[0,1]\to\mathbb{R}$ there is a Riemann-integrable function $g:[0,1]\to\mathbb{R}$ with ...
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1answer
51 views

Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$

Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate $$ \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx. $$ Well, the case $n=1$ is easy and the integral equals ...
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1answer
160 views

Bounds on integral

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where ...
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1answer
501 views

Normalization parameter, properties of Dirac delta functions

Suppose $\psi_E (x)=N(E)\exp (ikx)$ where $\psi_E (x)$ is a momentum eigenfunction, $N(E)$ is the normalization constant on the energy scale such that $\langle E'|E\rangle=\int_{-\infty}^\infty ...
3
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1answer
105 views

Evaluating $\int_{|z|=1} \sin\left(e^{\frac{1}{z}}\right) \ dz$

Let $$\int_{|z|=1} \sin\left(e^{\frac{1}{z}}\right) dz.$$ Is there an alternative to the residue theorem if we want to calculate the above integral?
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votes
1answer
141 views

Cauchy sequences when $p$ is a function

Consider $p$ being a positive bounded and measurable function and $\{f_k\}$ a sequence satisfying $$\int_{R^d} |f_k(x)|^{p(x)}dx<\infty$$ and $$\lim_{m,j\to \infty}\int_{R^d} ...
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1answer
187 views

Analytic expression for the primitive of square root of a quadratic

Can an analytic expression be given for $$\int \sqrt{ax^2 + bx +c} \, dx$$ I think substitution doesn't work in this case (I need to compute the integral $\int_0^t \ldots$).
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1answer
190 views

Is there any body of knowledge or study of the fractional calculus on definite integrals?

The fractional calculus is partly about nested indefinite integrals. Is there any study or body of knowledge on nested DEFINITE integrals? For example, the fractional calculus helps with this ...
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1answer
167 views

Order of integration

I am reading a book by L. D. Landau titled Mechanics and there is a "changing order of the integral" step on page 28 that I don't get: $$\int_0^a\int_0^E \left[{dx_2\over dU}-{dx_1\over ...
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2answers
521 views

What are your favorite integration tricks?

I'm learning to integrate and I'd like to hear what are you favorite integration tricks? I can't contribute much to this thread, but I like the fact that: $$\int_{-a}^{a}{f(x)}dx=0 ...
3
votes
2answers
209 views

Finding a closed-form solution to a definite integral.

I want to find a symbolic expression for the following integral as a function of $f$ and $g$: $$ \int_{0}^{\pi} \sqrt{1-\frac{2}{f + g \cdot \cos \theta}} \, d\theta $$ It is guaranteed that $f$ and ...
3
votes
1answer
146 views

Constant Radon-Nikodym derivative

Let $(\Omega, F, \mu)$ be a complete measure space, $\mu(\Omega)=1$ and $\mu$ takes values 0 or 1.Let $\nu$ positive measura, $\sigma$-finite and absolutely continuous with respect to $\mu$. Show ...