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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Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx ...
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54 views

Prove an equation about fractional integral

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Can anyone help me solve this? I really appreciate. If $f$ is continuous on $[0, \infty)$, for $\alpha \gt ...
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31 views

Calculus: Reduction formula

For this question, I can find out $I3$, but I have no idea how to find the reduction formula. Please advise me.
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47 views

Question about integral

I want to prove the inversion theorem on $R^{k}$.Then I need to compute the integral : $\int_{R^{1}}\exp^{-\sqrt{x^{2}+a^{2}}+itx}dx$ I have no ideal how to deal with it.I will appreciate your help. ...
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42 views

How to estimate this special integral?

Let $\theta\in(0,\pi)$, and $$ {\rm I}\left(\lambda,\theta\right) = \int_{\theta}^{2\pi - \theta} \left[({1 \over \sin\left(t\right)}\frac{\partial}{\partial t})^{k}{\rm e}^{\left({\rm i}\lambda - ...
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23 views

Stationary Phase approximtion for this type of integral

I would like to approximate the following integral for large $t$ : $I(t)=\int_0^{\pi}dx f(x)e^{iS(x)t}$ $S$ is real and $S'(0)=S'(\pi)=0$. $f$ is real and $f(0)=f(\pi)=0$ and $f'(0)=f'(\pi)=0$. ...
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31 views

Proof of a Proposition

I am having trouble trying to proof a proposition that appears in a paper. It begins with ...
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105 views

Attempted proof of the first part of the Fundamental Theorem of Calculus

So I was trying to prove the first part of the Fundamental Theorem of Calculus in a different way using the Riemann sum definition of the definite integral, rather than the way it was presented in my ...
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91 views

Transforming a Riemann-Stieltjes integral related to the factorial

I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$ but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary ...
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67 views

Name of theorem?

I am trying to understand a proof which uses the following statement without further explanation, so I am wondering if this is a well known theorem? For the unit ball $B$ with radius $r>0$ and the ...
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91 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow ...
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40 views

Question about finding volume using integration?

The question is: Find the volume of the solid whose base is a circle $x^2 + y^2 = 81$ and the cross sections perpendicular to the $x-axis$ are triangles whose height and base are equal. Now what the ...
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38 views

Need help evaluating $\lim\limits_{n \to \infty} \frac{1}{n} \int_1^n \Vert\frac{n}{x}\Vert dx$

$$ \mbox{Evaluate}\quad \lim_{n \to \infty}{1 \over n}\int_{1}^{n}\left\Vert\,n \over x\,\right\Vert \,{\rm d}x $$ Where $\left\vert\left\vert\, x\,\right\vert\right\vert : \mathbb{R} \to \mathbb{R}$ ...
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73 views

Integrating With Respect To $x$

Suppose I have the first derivative of the function $y$, $\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function $y$ by integrating with respect to $x$: ...
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120 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
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43 views

calculating sum of a limit of integral

I am trying to calculate the following expression $$ \sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ...
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41 views

comparison of two integrals

Let $n \in N$. How to compare two integrals: $$ I_1=\int_0^{\infty}\left(\frac{\sin t}{t}\right)^n dt \quad \text{and} \quad I_2=\int_0^{\pi}\left(\frac{\sin t}{t}\right)^n dt\,\, ? $$ I've beet ...
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29 views

Integral computation - what's going on?

Let $\lambda >0$ and denote $$ \lim _{\varepsilon \to 0+} \frac{1}{|\xi |^2 - (\lambda + i\varepsilon )^2} = \frac{1}{|\xi |^2 - (\lambda + i0)^2}. ...
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56 views

Area between two curves with a certain domain.

I am trying to find the area between two curves over a certain domain. The region of integration is between $xy=5$, $x=9-y^2$ and the lines $y=1$ and $y=2$. I have to show that this can be written as ...
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38 views

Compute an indefinite integral of logarithms.

Is there a simple way of proving the following identity: \begin{eqnarray} \int \log(x) \log(x^2 + (x + W)^2) dx = \\ 2 x + \left(1 - \log(x)\right) \left(\log(e^{-\pi/2} W) W + 2 x + W ...
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45 views

Integral containing inverse $\tanh$

How can I solve this integral containing inverse $\tanh$? Does it have any antiderivative? $$ \int \cfrac{t^2}{\sqrt{r^2-t^2}} \cdot \operatorname{arctanh}\sqrt{1-t^2}\; \mathrm{dt} $$
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109 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
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87 views

Integration of figure whose base is a quarter circle not centered at origin using polar coordinates

How do I integrate $$ \int_{1}^{2}\int_0^{\sqrt{2x-x^{2}}}\frac{1}{\sqrt{x^2+y^2}}dydx $$ using polar coordinates? The base is a quarter circle of radius 1 centered at (1,0), so my first instinct ...
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52 views

Integration of infinite sum

I have a signal that I want to sample using delta functions. The signal is: $x(t) = W^2sinc^2(Wt)$ and after the sampling we will have the signal $z(t)$. We know the form of the signal in the ...
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204 views

Integrating a complicated function

After spending a couple of weeks, I was able to find the solution to a certain differential equation, given below (Well they are the eigenfunctions to be exact): $$y_n(x) = ...
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68 views

Formula for integral over hypersurface??

Can someone give me a formula for an (Lebesgue) integral of a function $f:M \to \mathbb{R}$ where $M$ is a bounded $C^k$-hypersurface of dimension $(n-1)$ in $\mathbb{R}^n$? I have tried the ...
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127 views

What is an example of an integral that CANNOT be done without contour integration ? If that exist.

What is an example of an indefinite integral that CANNOT be done without contour integration ? If that exist. Im talking about closed forms for integrals, not numerical methods. Note that there are ...
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66 views

Strategies for swapping the order of integration with dependent bounds

What are the general strategies for swapping the order of integration given dependent bounds? Specifically, in $\mathbb{R}^2$, Fubini's theorem allows us the following $$ \int_{a}^b\int_{c}^d ...
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35 views

How to calculate the following 3D ${\bf k}$-space integral?

I'm struggling to calculate$$ \sum_{a,b=\pm}\int\frac{\text d\mathbf{k}}{(2\pi)^3} ...
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287 views

Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
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46 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
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47 views

Help computing integral

I've been desperately trying to solve the following integral without much success. $$I(u)=\int_1^u \frac{e^{-x} (2 x-1)}{\sqrt{x~(A~e^{-x}+1)-B \sqrt{x}}}dx,$$ where $A,B\in \mathbb{R}$ are constants ...
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63 views

Show that $\left| \oint_{\partial D}fdx + gdy \right|^2 \leq (\text{Area}(D))\int_D \left( |\nabla f|^2 + |\nabla g|^2 \right) dx dy.$

Let $\vec{F}=(f,g):\mathbb{R}^2\to \mathbb{R}^2$ be a smooth vector field such that $|\vec{F}(x,y)|\to 0$ rapidly as $|(x,y)|\to \infty$, and let $D$ denote a compact domain in $\mathbb{R}^2$ ...
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37 views

On Daniell integral and the notion of measurability

Recently, I decided to learn Daniell integration and after a couple of months on it I like to think that I got the notions right. I also understood the Daniell-Stone theorem that established that, ...
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106 views

A problem on Riemann Stieltjes Integral

$ \int_{0}^2 x\,d \alpha $ where $ \alpha (x) = x $ if $ 0\le x\le 1 $ and $ \alpha(x)=2+x $ when $ 1<x\le 2 $ I did this by taking a partition which divided the interval $[0,2]$ to $2n$ equal ...
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56 views

Integrability and product measure

Let $X$ and $Y$ be subsets of $\mathbb{R}$, and let $\mu$ be a measure on $X$ and $\nu$ a measure on $Y$. Let $f : X \times Y \rightarrow \mathbb{R}$ be $\mu$-summable and $\nu$-summable, i.e. ...
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29 views

Evaluate spatial variation of density-like scalar

Apologies if this has been asked previously, but I'm not totally sure of the best way to pose the question. Background I'm evaluating the variation of a spatially varying scalar field $p$ ...
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64 views

There exists $c\in [a,b]$ such that $\int_a^c f(t)dt = \int_c^b f(t)dt$

If $f:[a,b]\longrightarrow \mathbf{R}$ is integrable prove that there is $c\in[a,b]$ such that $\int_a^c f(t)dt = \int_c^b f(t)dt$. I set $g(x)=\int_a^x f(t)dt$ but I don't know how I must continue.
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53 views

Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
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50 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
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68 views

Proving the converse of the Cauchy criterion for integration

Prove the converse of the Cauchy criterion for integration. That is, prove that if $f$ is integrable on $[a,b]$, then for any $\epsilon>0$ there is a $\delta>0$ so that for any partitions ...
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33 views

Inequality: double jensen, is this correct?

Imagine I have an exponent $a=bc$ where $b>1$ and $0<c<1$, the product $a=bc\in \mathbb{R}$ might be greater, equal or less than 1. Then for a measurable set with measure 1. Can I say the ...
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161 views

Find the volume of the region bounded by $z=x^3 + y^2 $, $z=0$, $-a<x<a$, and $-a<y<a$

Since $z \geq 0$ then, $x^3+y^2 \geq 0 \rightarrow x \geq -y^{2/3}$ So, I set up the integral as follows: $$V = \int_{-a}^{a} \int_{-y^{2/3}}^{a} (x^3+y^2) \, dx \, dy$$ However, according to ...
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73 views

Ring Integration

In thinking about various methods of integration, I began to wonder if there was some sort of unifying theory relating integration and ring theory. For example, would there be a way to make sense of ...
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60 views

Evaluating $\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h_1^2}dx\int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h_2^2}dx$.

How to evaluate the integral $$\displaystyle\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h^2}dx \int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h^2}dx$$ and ...
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72 views

How to compute the arc length of $f(x) = ax + b \sin(x)$.

I would like to compute the length of the arc of $f(x) = a x + b \sin(x)$ (let's say from $0$ to $\alpha < 2\pi$.). The traditional method of computing it as the integral $\int_0^\alpha ...
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80 views

Numerical Methods for estimating divergence over an improper integral

Problem given a function $f(x)$, defined on $[ \epsilon, \infty )$. Is there a way to "numerically estimate" whether the integral of the function diverges over the domain $[ \epsilon, \infty )$? ...
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34 views

Riemann integrability in not sequentially complete LCS?

For $E$ any Hausdorff locally convex space, I have been wondering whether Riemann integrability of all continuous functions $f:[\,0,1\,]\to E$ implies that $E$ be sequentially complete. For example, ...
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17 views

Compute tetrahedral region

Show that the volume of region $A$ is $1/6$. Region $A$ is a tetrahedral region in $\mathbb R^3$. $$A=\{(x,y,z)∈R^3 \mid x\ge 0, y\ge 0, z\ge 0, \text{ and } x+y+z\le 1\}$$
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80 views

how to solve this indefinite integral?

could anyone help me how to solve this indefinite integral? $\int{dx\over \sqrt{\sin^3 x+\sin (x+\alpha)}}$ Thank You.