# Questions tagged [integration]

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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### Evaluate a pair of integrals involving dilogarithms over the unit interval

These are two variations on the "Bonus round" problem, expertly address by student at the end of his answer to A pair of integrals involving square roots and inverse trigonometric functions over the ...

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**4**answers

405 views

### Closed-form expression for certain product

$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...

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35 views

### How to find the volume using triple integral spherical coordinates? [on hold]

A hemispherical bowl of radius $5 cm$ is filled with water to within $3cm$ of the top. Find the volume of water in the bowl?
How can I find the volume using spherical coordinates?
writing it like ...

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**1**answer

184 views

### A pair of integrals involving square roots and inverse trigonometric functions over the unit disk

I would like to perform the integration ($u \in [0,1]$),
\begin{equation}
\int_{a=-1}^1 \int_{b=-\sqrt{1-a^2}}^{\sqrt{1-a^2}} u^2 \sqrt{-a^2 u^2-b^2+1} \tan ^{-1}\left(\frac{\left| a\right|
}{\...

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**1**answer

90 views

### A question about integration of spherical harmonics on $(S ^ 2, can)$

Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that
$$ \int_{\mathbb{...

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**0**answers

54 views

### Reference for calculating definite integral involving sines

Recently I accidentally discovered a simple, elementary derivation of the following identity, valid for any $n,k \in \mathbb N$:
\begin{align*}
\frac1\pi \int_0^\pi {\rm d}x \left(\!\frac{\sin nx}{\...

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**0**answers

48 views

### Numerical expectation involving Dirac-delta function

I'm looking for the way for numerical integration including Dirac-delta function. Here is what I want to obtain in numerical way such as Monte Carlo sampling.
$$ \int m(\mathbf{x})\delta(G(\mathbf{x}))...

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vote

**2**answers

140 views

### Integral formula involving Legendre polynomial

I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...

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51 views

### How to prove the binary function uniformly boundary?

Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2.$ For $x,y>0,$ define a fucntion
$$H(u,v)=\frac{u^{\frac{1}{2}}\int_0^{\infty}\int_0^{\infty}\...

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**3**answers

251 views

### Computing the volume of a simplex-like object with constraints

For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...

**2**

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**0**answers

238 views

### Integration over a Surface without using Partition of Unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...

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**0**answers

13 views

### Can you add an extra e^x when integrating? [migrated]

So I've been given this problem to solve (pretend it's a fraction or click the link to see the question please)
∫ -26e^x-144
e^(2x) + 13e^x + 36
problem ...

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**3**answers

227 views

### Integration for Dirac-delta function

Is there any way to solve the integration below? or make it simple to eliminate the Dirac-delta function?
$$\int_{-\infty}^\infty m(x)\delta(G(x)-g_c)f_X(x)dx $$
where $f_X(x)$ is a probability ...

**11**

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**1**answer

341 views

### Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...

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**1**answer

76 views

### “Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇”

This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...