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      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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      4answers
      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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      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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      1answer
      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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      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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      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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      2answers
      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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      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 \...
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      0answers
      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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      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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      3answers
      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 ...
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      1answer
      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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      1answer
      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 ...

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