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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94 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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38 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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34 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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27 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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31 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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34 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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41 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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83 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 ...
2
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64 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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42 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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92 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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32 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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30 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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112 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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41 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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44 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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42 views

Why is the equality true: $\int_{u=0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, du =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$

I have the following equality in a textbook of mine $$\frac{y^{\alpha_1+\alpha_2-1} e^{-y/\beta}}{\Gamma(\alpha_1+\alpha_2) \beta^{\alpha_1+\alpha_2}} \cdot ...
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53 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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64 views

Closed form for $\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$

Find a closed form for $$\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$$ What I have tried Expanding the $\mathrm{arccsch}$ into its logarithmic form, however I ...
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64 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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34 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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27 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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56 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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83 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
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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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43 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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56 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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111 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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57 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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56 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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45 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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32 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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15 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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62 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.
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125 views

Proving $\int_a^b \frac {x dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} $

Can anyboby help me? How to prove that $$ \int_a^b \frac {x dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} $$
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54 views

Approximating a function using its integral

Question: Let $f:\Bbb R \to \Bbb R \in C^{1}, \forall \delta>0:$ $$F_\delta = \frac 1{2\delta}\int^{x+\delta}_{x-\delta} f(t) \, d(t)$$ in $[a,b]$ prove that $\forall \varepsilon>0 \exists ...
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53 views

$\int \dfrac {xe^{2x}}{1+2x^2}~dx$

Working on integration by parts problem set. Stuck with this problem. $\int \dfrac {xe^{2x}}{1+2x^2}~dx$ Tried the following: Let $u=e^{2x}$, therefore $du=2e^{2x}~dx$ $dv=\dfrac ...
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84 views

Is maxima able to solve such integral? If no, what can?

I was sitting and decided to get a precise magnetic field equation from Bio-Savar law, and got out this handsome formula (by the way, is there a way to use ASCIIMath here on Mathematics?): $$ ...
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53 views

Tricky Integral Involve Sine

How would I integrate the following: $$ \int \frac{c}{\sin(t)\sqrt{\sin^2(t) - c^2}} \, dt $$ Here $c$ is a constant. I have tried numerous substitutions, but I just can't seem to get the right one. ...
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218 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
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55 views

Two-dimension recursion formula for computing volumes

The two-dimension recursion formula for computing volumes of balls says: A proof of the recursion formula relating the volume of the $n$-ball and an $(n -2)$-ball can be given using the ...
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89 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
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22 views

Continuous weighted means

Let $f$ and $\omega$ be a positive functions over $(a,b)$ ( $f:(a,b)\to\mathbb R^+$, and $\omega:(a,b)\to\mathbb R^{\ge0}$ ) such that $$\int_a^b\omega(x)dx=1$$ Under which conditions can I guaranty ...
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29 views

Integral to be computed

I am interested in computing $$J(a,b):=\int_{\mathcal{I}(a,b)}\frac{dx \ dy\ dz}{|ax^3+ay^2+bz^3|^{2/3}}, $$ where $a,b$ are natural numbers and $$\mathcal{I}(a,b):=(0,1]^3\cap\{x,y,z \in ...
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271 views

Reduction formula for a trigonometric integral

I have come upon the following trigonometric integral: $$\int (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x,$$ where $\alpha \in \mathbb{R}$ is an arbitrary real constant and $n \in \mathbb{N}$ is a ...
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39 views

Can every bounded function on an interval be “integrated”?

There are of course various and increasingly general notions of integrability, including some notions on the real line that I'm told properly generalize Lebesgue integration in that context. Is one ...
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85 views

An example of Risch algorithm for integrating $y$, $F(x,y)=0$.

I would like to compute the integral $$ \int y\,dx, \qquad y=\sqrt{x+\sqrt{x+1}},\\ F(x,y)=y^4-2xy^2+x^2-x-1=0, $$ in closed form, where $F(x,y)$ is a polynomial in $\mathbb{C}[x,y]$. I am trying to ...
2
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59 views

Evaluating an integral $\int \sqrt{x^{2k}+x^{-2k} - 1} dx$ (encountered in pursuit problem)

Evaluate the indefinite integral $$\int \sqrt{x^{2k}+x^{-2k}-1} dx$$ where $k\in \mathbb{R}$. Source of inspiration: The pursuit problem of fox on rabbit. Rabbit with speed $v_R$ starts from origin ...
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85 views

Triple Integral $\iiint_D(x^2+y^2)dV$

Suppose $D$ is minimum of the two solid regions limited by the two surfaces $x^2+y^2+z^2=6a^2$ and $z=\frac{1}{a}(x^2+y^2)$,($a$ is a positive constant). Please help me to compute the value of this ...
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88 views

Evaluating integral with branch of log

I'm having trouble understanding what this question even means really. "Let $\gamma$ be the semi-circle from $2i$ to $-2i$ that passes through $2$ in the positive direction. Find $\int_\gamma ...