Questions tagged [approximate-integration]

Use this tag for questions related to approximate integration, which constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

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19 views

Approximation of a function using n intervals

Consider the function 𝑓 𝑥 = 𝑠𝑖𝑛 𝑛𝜋𝑥 and divide the domain 0 ≤ 𝑥 ≤ 1 into m intervals. For the “exact” approximation of the function, we will use 𝑚 = 100 intervals. Plot the “exact” function ...
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19 views

Numerical quadrature with preassigned points

I have been looking for a numerical quadrature that might be possible to pre-assign specific nodes. For instance, I need to numerically calculate the integral of $f(x)$ in the interval $[a,b]$ but the ...
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1answer
14 views

Spectrally-Accurate Quadrature of Singular Integrand

I have a set of PDEs governing some function $f(r)$ which I desire to solve via a psuedospectral method (we can consider $f$ to be smooth). It is defined on the interval $r\in[0,\infty)$ with symmetry ...
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22 views

An integral approximation for $Q = \sum_x f(x) $ when $\tan(C x) = x$

I have a sum of the form $$Q = \sum_{x}f(x).$$ Here, the $x$ entering the sum are the countably infinite set of solutions to $$ \tan(C x) = x.$$ The $x$ are not integers, but they do satisfy $$ \frac{\...
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16 views

How to determine error value needed to attain certain decimal precision with numerical integration

I need to use numerical integration to integrate a function $f(x)$ from $a$ to $b$ correct to $k$ decimal places. The two methods I am interested in are the Trapezoidal rule and Simpson's rule. I'm ...
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1answer
21 views

Numerical Integration on Some Points in xy-plane

I have an assignment to finding the result of the integration of some points given in the closed interval $[0,1.2]$ with the method of Trapezoidal Rule, Simpson 1/3 and Simpson 3/8 rule. The problem ...
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111 views

To determine the integration of $ \int_{0}^{+\infty} \exp\!\Big(-\Big(\frac{ax^2+bx+c}{gx+h}\Big)\Big) dx$.

What is the integration of the following function: $$ \int_\nolimits{0}^{+\infty} \exp\!\bigg(-\bigg(\frac{ax^2+bx+c}{gx+h}\bigg) \bigg)dx.$$ What I have done is as follows: Here, $\kappa=c-\Big(\...
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58 views

LaGrange Multiplier constrain with discretised functions (Matlab)

I am a little stuck with my optimization assignment. It's, no surprise, the catenary but the approach we are looking for is a little different than whatever I could track down on different forums ...
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1answer
30 views

Problem understanding the Stationary-Phase-Approximation example of Wikipedia?

I am trying to understand the Stationary-Phase Approximation from the Wikipedia example showed in here, but there is something I don´t understand how is made work. Following the article's example: $$f(...
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1answer
74 views

Please help me solve this integral [closed]

Please help me solve this integral \begin{align} \int_{-\pi/2}^{\pi/2}\sqrt{\varepsilon_r-\sin^2\varphi}\cos^2\varphi\ d\varphi. \end{align} Thank you very much!
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The time complexity of integral evaluation

I have the following integral: $$p(x) = \int p(\pmb{\mu})\prod_{i=1}^n\sum_{c_i}p(c_i)p(x_i|c_i,\pmb{\mu})d\pmb{\mu}$$ $\pmb{\mu} \in \mathbf{R}^K$ . The time complexity to numerically evaluate this ...
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21 views

Area integral over a function without analytical form

Given a parametric surface $S(u, v): \mathbb{R}^2 \to \mathbb{R}^3$ ($0 \le u,v \le 1$) and an implicit function $f(x,y,z): \mathbb{R}^3 \to \mathbb{R}$, find the integral of $f(\cdot)$ over points on ...
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82 views

Integral of the reciprocal of a sum of complex exponentials

I am interested in (realistically, a closed form approximation to) the (real and imaginary parts of the) following integral: $\int_I \frac{1}{\sum_{s=1}^S e^{i a_s x + b_s}} dx$ where $x$ is real and ...
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62 views

An integration involving Sinc function

Sir, I have been trying to find out the time average of the Intensity distribution of a scattering problem but I could not find the desired closed form answer of the following integral, which shows ...
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170 views

Approximating the integral of exponential of hyperbolic functions

Let us consider the following integral $$ I(x) = \int_{0}^{\infty} e^{-x \cosh{\theta}} d\theta $$ where $x \in \mathbb{R}$. I am trying to get an approximation to this integral for the regime where $...
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58 views

Approximation of a complex integral

Given the complex integral $$I(y) = \int_{-\infty}^{+\infty}\left(\exp(-\mathbf{i}2\pi u)\frac{h\left(\frac{u}{d}+y \right)-h\left(\frac{u}{c}+y \right)}{2\pi u} \right)du \qquad \text{with } y \...
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99 views

For help an inequality that for large $n$, whether $\frac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$ for some $\beta>0$

Can anyone help me to prove that for large $n$ whether $$\dfrac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$$ for some $\beta>0$ where and $\big[x\big]$ denotes the ...
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103 views

Ask for help a inequality problem, does $\sum\limits_{i=0}^{k-2}\log_2\left(\frac{n-i}{k-i-1}\right)>c\cdot n$ [closed]

Can anyone help me to give me a detailed proof (or disproof) of the following $\sum\limits_{i=0}^{k-2}\log_2\left(\frac{n-i}{k-i-1}\right)>c\cdot n$ for some constant $c>0$, where $k=\Big[\frac{...
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1answer
75 views

Deriving Gauss-Hermite weights

I'm currently dealing with Gauss quadrature and I'm having trouble deriving the formula for the Gauss-Hermite quadrature weights. For reference: in my course the Hermite polynomials are defined with ...
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251 views

Approximation of integral of gaussian function over a parallelepiped

Given a multi-dimensional gaussian function, defined by $$f(\boldsymbol{x})=\exp\left\{-\frac{1}{2} \boldsymbol{x}^T\boldsymbol{x} \right\}=\exp\left\{-\frac{1}{2} \sum_{i=1}^nx_i^2 \right\}$$ And an ...
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1answer
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Given $n$ subsections, how many parabolas are made in Simpson's rule?

Here's a picture from Stewart's calculus for Simpson's rule Since a parabola goes through $x_n, x_{n+1}, x_{n+2}$, we would get $\frac{n}{2}$ parabolas? So in this case of $n=6$, then we'd get $3$ ...
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72 views

Value of integral greater than $\frac{\pi}{2}$

Is there an elementary way to prove that $$\int\limits_{-1}^1 \frac{\arccos x}{\exp(x)+1}>\frac{\pi}{2} ~~?$$ The approximate value of the integral is 1.75. I have tried to restrict to a smaller ...
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55 views

Numerical integration of samples of a function

I have a function $c ( I (\vec{r}) )$. Not a constant, c doesn't denote a constant. So $c$ is a function of $I$ which is a function of $\vec{r}$. This is hard to sample and I have sampled it for 10,...
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64 views

How to evaluate given integral in terms of Gamma function?

I want to evaluate the below integral $$ \int_{0}^{\infty} e^{-ax^{2}-bx} x^{\alpha -1}\,\mathrm{d}x, ~~~~~~ \operatorname{Re}(\alpha)>0 $$ where $a,b$ are real constants. Somehow, I want to ...
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1answer
60 views

How to approximate integral resulting from physics problem?

Setup I'm taking a physics class and need to calculate (or approximate) a particular integral. Let $$\Delta(x):= 1-(1-x)x \frac{p^2}{m^2},$$ where $p^2:= p_\mu p_\nu\eta^{\mu\nu}$ with $\eta$ being ...
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191 views

Some of approximations related to integrals and elliptic functions

I have just known about the elliptic functions and I saw three nice examples as following : $$\int_{0}^{\frac{\pi}{2}}\frac{\tan x}{4\ln^{2}\tan x+ \pi^{2}}{\rm d}x\cong\frac{1}{4}$$ $$\int_{0}^{1}\...
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Boole's Rule on reading gyroscope data.

So I've read this paper about gyroscope and Boole's rule and it said they got orientation of a device by calculating Boole's integral. I do have basic understanding about Boole's rule, but I have no ...
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2answers
95 views

How to approximate the integral $\int_{\mu}^{\infty} \frac{\sqrt{x^2-\mu^2}}{e^x-1} dx$ for a constant $\mu \gg 1$

I think this integral has no exact solution, but I've found that you can approximate it for large $\mu$ as: $$\int_{\mu}^{\infty} \frac{\sqrt{x^2-\mu^2}}{e^x-1} dx \approx \sqrt{\frac{\pi \mu}{2}} e^{-...
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62 views

solving $\int_{-\infty}^\infty \frac{\alpha y\cdot e^{-\alpha\sqrt{x^2 + y^2}}}{\sqrt{x^2 + y^2}}dx $

Trying to solve the following integral, $$\int_{-\infty}^\infty \frac{\alpha y\cdot e^{-\alpha\sqrt{x^2 + y^2}}}{\sqrt{x^2 + y^2}}dx $$ where $\alpha$ is a constant. this is an integral over $dx$ of ...
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error bound of $I=\int_{0}^{1}\left(f'(x)\right)^{2}dx\thickapprox h\sum_{i=1}^{N}\left(\frac{f(ih)-f((i-1)h)}{h}\right)^{2}$

let for $f\in c^{3}$ and $I=\int_{0}^{1}\left(f'(x)\right)^{2}dx\thickapprox h\sum_{i=1}^{N}\left(\frac{f(ih)-f((i-1)h)}{h}\right)^{2}$. whereas $h=\frac{1}{N}$ find the maximal error bound in [0,1]. ...
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64 views

Approximations to High Dimensional Integrals

Is there a general method for analytically approximating integrals of the form $$ \int_{\mathbb{R}^n} d^{n}\textbf{x} \, \exp(- F_n(\textbf{x})),$$ where $\textbf{x} = (x_1, x_2, \ldots, x_n)$, $F_n(\...
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1answer
52 views

Fresnel integral: stationary phase approx.

The following integral describes the propagation of light (in certain cases) $$ U_i(x,y,z) = \frac{e^{ikz}}{i\lambda z} \int_{-\infty}^\infty d\xi \int_{-\infty}^\infty d\eta \; U_0(\xi, \eta,0) \, ...
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1answer
58 views

Expanding convolution integrals for sharply-peaked functions.

Question: I have integrals like $$ I(x) = \int_0^\infty K(y) F(x,y) dy $$ where $K(y)$ is some kernel function that is sharply peaked at some special value $y = y_0$ and otherwise trends toward zero ...
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335 views

How to approximate a double integral numerically using the Simpson's Rule

.I'm trying to approximate the integral : $$\int_{0}^{1} \int_{0}^{1} \ln(e^x+e^y+x+y)\, dx\,dy$$ with an error $\leq 10^{-4}$. I tried to use the Simpson's rule, and trapezoidal rule. But I had ...
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44 views

Struggling to evaluate an indefinite integral

I tried solving the following integral $$\int \frac{dx}{(1-\frac{c}x{} ) (\frac{3c}{x}-1)^{\frac{1}{2}}} $$ How could I incorporate the approximation that x $\approx$ c as I am told that would ...
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1answer
40 views

Finding largest $r$ and smallest $q$ in $\mathbb{N}$ so that $\exp(r)\lt\int_{1}^{2}\exp(x^3+x)\mathrm dx\lt \exp(q)$ by hand

Initially I was given that $k_{1}\le \int_{1}^{2}\exp(x^3+x)\mathrm dx\le k_{2}$, with few options of what $k_{1},k_{2}$ could be. With the very basic technique of approximation, one can argue that ...
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1answer
28 views

Approximating function by its step approximation

$f$ be any bounded measurable function on $[0,1]$. Define the step function $f_n(x)=f(\frac{i}{n})$, if $x \in [\frac{i-1}{n},\frac{i}{n}]$. Is it always true that $$\int |f(x)- f_n(x)|dx \rightarrow ...
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34 views

Integrating over roots

(I'm an independent learner, so no professor to ask.) There is a short example given in a youtube video not in given in any textbooks that I don't really follow. Yet it seems very useful, and possibly ...
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1answer
138 views

Numeric calculation of infinite Fourier integral in 2D

Consider a 2D function $f(x,y)$ on $\mathbb{R}^2$, which is finite and decays on some finite interval. I don't have a nice analytical/closed-form expression for $f(x,y)$, but can evaluate it at any $(...
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1answer
78 views

Laplace's method for integration

I am trying to compute $$I(\lambda) = \int_{0}^1 \frac{x}{\sqrt{1+x^4}}e^{\lambda x}\mathrm{d}x $$ for large, real, positive $\lambda$. I'm attempting this with Laplace's method as suggested, however ...
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1answer
127 views

Asymptotic expansion of integral of airy function

In this question I am given that the asymptotic expansion of the Airy function for large $z$ is given by $$Ai(z) = \frac{1}{2}\pi^{-\frac{1}{2}}z^{-1/4}\exp\left(-\frac{2}{3}z^{\frac{3}{2}}\right)\...
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approximation of $\int \ln(1+\frac{2k}{a+b-k}) $

I'm trying to calculate an integration over an arbitrary polygon $P$, and a point $p$ is on $P$. $k$ is a constant, and $A(x,y,z)$ and $B(x,y,z)$ are arbitrary functions returning a scalar. As $\int_P ...
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77 views

Estimate the error in the trapezoidal rule

I have the following exercise: We numerically approximate the integral $\int_{x_{L}}^{x_{R}}f(x)dx$ with the trapezoidal rule, using $x_{L}$ and $x_{R}$ as the only nodes. Next, we halve the interval $...
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261 views

Why is the Gaussian-Legendre Quadrature so effective?

I understand how it works, how its derived, etc. The proof of it has been shown to me. That is to say, I know how Legendre polynomials are derived, I know they are orthogonal, I know we sample a ...
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63 views

Piecewise Gauss-Legendre quadrature order of convergence

Given a definite integral $\int_a^bf$, If we increase the number of nodes and weights of the Gaussian quadrature, we would get closer to the exact integral. But another way to get more exact is to ...
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48 views

Romberg Integration: What are the rules to using it, as my implementation fails?

Im attempting to numerically evaluate $\int\limits_{-1}^1 \cos^2(x) dx$. I know the exact answer ahead of time: $1 + \cos(1)\sin(1)\approx 1.4546487134\ldots$. Im doing this in Microsoft Excel, to ...
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1answer
55 views

Simpson's Rule Question with Example

I'm just learning Simpson's Rule for integral approximation and I have a question. $ \int _a^b\:f\left(x\right)dx\:approx=\frac{\frac{b-a}{n}}{3}\left[\left(1f\left(x0\right)\right)+4f\left(x1\right)+...
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30 views

Solution or approximation of $\int_0^r exp(-g*\sqrt{x^2+2*b*x+c})dx$

I'm trying to find a solution or approximation of $$\int_0^r \exp(-g\sqrt{x^2+2bx+c})dx$$ where g > 0, c > 0 and r is from 0 to infinity. I found that $$\frac{1.5}{b*\sqrt{c+1.5}+c+1.5}$$ Works ...
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1answer
37 views

Approximate the same integral on different range of same length

Let $f(x)=(\tan x)^{\frac{3}{2}}-3\tan x+\sqrt{\tan x}.$ Consider the three integrals $$ I_1=\int_0^1f(x)\ dx, I_2=\int_{0.3}^{1.3}f(x)\ dx, I_3=\int_{0.5}^{1.5}f(x)\ dx $$ Then how to show that $I_1&...
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4answers
98 views

Approximating the integral $\int_1^\frac{3}{2}\frac{\ln(x+1)-\ln(2)}{x-1}dx$ to accuracy of $0.001$

I need to approximate the integral $\int\limits_1^\frac{3}{2}\frac{\ln(x+1)-\ln(2)}{x-1}dx$ with an accuracy of $0.001.$ How don't know how I can do it, since the integrand doesn't have an elementary ...

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