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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33 views

Approximation for the Marcum Q-function

I want to approximate the Marcum Q-function $Q_{m}(a,b)$ when $b$ goes to zero, where the Marcum Q-function $Q_{m}(a,b)$ is defined in an integral form, shown in https://en.wikipedia.org/wiki/Marcum_Q-...
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0answers
12 views

Composite trapezoidal quadrature error convergence

Let $\hat{T_n}(f)$ be the composite trapezoidal quadrature for the integral $I(f)=\int_{a}^{b}f(x)dx$ based on $n$ equal subintervals of $[a,b]$. We define: $$\hat{Q_n}(f)=\hat{T_n}(f)- \frac{(b-a)^...
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0answers
47 views

Indefinite Integration of real valued function [duplicate]

Why $\int e^{-x^2} dx$ cannot be expressed in a closed form? Definite integration of this function is possible however I could not understand why indefinite Integration is not possible.
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3answers
131 views

Approximating $\int_{-\infty}^\infty \frac{e^{-y^2/2}}{((y+y_0)^2+x_0^2)^r}\,dy$

I have to estimate the integral $$\int_{-\infty}^\infty \frac{e^{-y^2/2}}{((y+y_0)^2+x_0^2)^r} \,dy,$$ for $r\in \mathbb{R}^+$. I am a little amazed that Sage and Wolfram Alpha have nothing to say ...
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2answers
49 views

Trapezium Rule Exam Question

Use the Trapezium rule to estimate the area between the curve $y = x^2 -8x + 18$ and the $x$ axis from $x = 2$ to $x = 6$. Use $4$ strips of equal width. What I did: height $= \frac{(b - a)}{n}$ $= \...
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1answer
20 views

Error Bound of composite trapezium rule

Given the function: $f(x) = \cos(2x) \exp\left(-x^2\right)$ I estimated $\int_{-2}^2 f(x) \ dx$ using the formula. I need to calculate the error bound using the formula: $$ R = −\frac{b−a}{12} \cdot ...
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0answers
31 views

Approximate a double integral

I am struggling to approximate the following integral $$\sqrt n\int_0^\infty \int_0^\infty (1 + n x^2)^{-1}(1 + y^2)^{-1} \Phi\left(\frac{a}{\sqrt{1 + b + x^2y^2}}\right) \text{d}x \text{d}y,$$ where ...
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0answers
33 views

How do you determine a step function to approximate another function?

I have a question here in which I need to explicitly write down a sequence $f_n(x)$ that can approximate $e^x$. From reading, I known that I need to pick a partition sequence of $x_k$ so that I can ...
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0answers
46 views

Fourier partial sums of Sawtooth wave are not equal its convolution with the Dirichlet kernel!

Let $f$ be the $2\pi$-periodic function relating \begin{equation} f(x) = \frac{\pi-x}{2} \end{equation} on $(0, 2\pi)$. The coefficients of its Fourier series are easily calculated [see (*), ...
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1answer
43 views

Integration involving hermite polynomials

The integration shown here appears in the stationary solution of the perturbed non linear oscillators in Physics. Is there any direct way to perform the definite integration of the form shown here $\...
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0answers
43 views

How can I solve this integral function?

I want to find set of possible equations for $a(t), b(t)$ and $B_o(t)$ for which I would be able to integrate this function. (Later I have to choose from the possible set of functions for which $P(t)$ ...
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1answer
17 views

Integrating with a non-analytical solution (random effects)

I would like to integrate a function with two random effects, implying three successive integrations. My problem is that after the first integration, it is not possible to obtain an analytical ...
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1answer
47 views

Error analysis of approximating Fourier transforms

Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ \hat{f}(k) = \int_{-\infty}^{\infty} dx~ f(x)~ e^{i k x} .$$ Suppose I want to approximate this transform by a ...
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1answer
39 views

Are there any numerical integration methods that do not involve rectangles or polynomial approximations?

For the Riemann integral, are there any methods of numerical integration that do not involve rectangles or approximating the area with a polynomial function? I am aware of the trapezoidal rule, but I ...
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1answer
39 views

Approximate definite integral near singularity of integrand

I would like to expand integrals of the form $$ - \int_1^{1-\varepsilon} \frac{f(x)dx}{\sqrt{1-x^2}} $$ where I know that $f(x)$ is well-behaved around $x=1$ but otherwise is a free function here (if ...
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0answers
16 views

Expected value of inverse of surrounding density

Fix a probability distribution on a compact set $\mathcal{X} \subset \mathbb{R}$. I wonder what the conditions would be such that $\mathbb{E} \left[ \frac{1}{F(x+d)-F(x-d)} \right]$ does not diverge ...
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1answer
28 views

Analytical approximation for logit-normal-binomial distribution

As I understand, there is no closed form expression for $$f(x, \mu, \sigma) = \int_0^1 p^{(x-1)}(1-p)^{n-x-1}\exp\left(-{(\text{logit}(p) -\mu)^2 \over 2\sigma^2}\right)dp.$$ Is it possible to ...
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0answers
122 views

References on quadrature of $\int_0^{\infty} f(x) \exp(-x^a) dx$

I am aware of Hermite-Gaussian quadrature techniques for integrals of the form $$ \int_0^{\infty} f(x) \exp(-x^2) dx $$ However, am I looking for references on quadrature where the exponent is more ...
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0answers
131 views

Integral of a harmonically phase-modulated sinusoid

I am trying to calculate integrals of the form $$ I_n = \int_0^1 \cos\left[2\pi nt + \sum_{k=1}^N \beta_{nk} \sin(2\pi kt) \right] \mathrm{d}t, $$ where $n \in \{1, \ldots, N\}$ for some positive ...
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0answers
30 views

Changing Integration measure from cartesian to polar introduces divergency

I stumbled over the following problem related to the Coulomb kernel that I hoped some of you might understand. I am interested in the function, that is given by the cartesian integral: $$f(x',y')=\...
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1answer
95 views

When to use which “closed” Newton Cotes rule?

Given a set of datapoints, I was thinking about when to use which (closed) Newton-Cotes formula? I developed a decision tree which would go like this: Are the given datapoints equally spaced? ...
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2answers
322 views

What is the order of the midpoint rule?

If the Trapezoidal-Rule has the order $n=1$, and Simpson's has order $n=2$, what is the order $n$ of the midpoint rule? And if the weights of the Trapeziumrule are ($1/2, 1/2$) and those of the ...
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1answer
29 views

COnverting integral into First Order of Bessel Fuuction of first kind

How to prove $$ \frac {\omega^2 \int_0 ^{2\pi/\Omega} \sin \left(\Omega s\right) \sin \left(A \cos \left(\Omega s\right) \right)ds}{\int_0 ^{2\pi/\Omega} A \sin \left(\Omega s \cos \left(\Omega s ...
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1answer
23 views

Matching the orders of numerical solvers.

Let's say I wanted to solve a system of ODEs using RK4, then I want to take the average value of one the solution components over some interval using some integration method like the trapezoid rule. I ...
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0answers
253 views

Simpson's Rule in Matlab [closed]

I have made the following code based on Simpson's expansion: function I = simprule(f, a, b, n) h = (b-a) / n; x = a:h:b; S = 0; L = 0; for l = 1:2:n %generates the odd number array S = S + 4*...
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2answers
44 views

Bounding a somewhat complicated integral (exponential of a polynomial)

I am interested in bounding the following integral, where $a>0$ is a constant: $$\int\limits_0^a \exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right) dx$$ I first conjectured that $$\int\...
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2answers
165 views

Error formula for Composite Trapezoidal Rule

My textbook gives me the error term for the Composite Trapezoidal Rule as this: $-\frac{b-a}{12}h^2f''(\mu)$, where $\mu \in(a,b)$ and $f \in C^2 [a,b]$ I am using MatLab to produce approximations ...
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1answer
35 views

Composite Lagrangian Quadrature rule for sin(x)

Suppose we want to estimate $\int_0^{h}f(x) dx$ using Lagrangian interpolating polynomials and with nodes $x_0 = 0,x_1 = \frac{2}{3}h$. Thus we compute the quadrature rule $$ Q(f) = a_0f(0)+a_1f(\...
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1answer
33 views

Definite integration (approximate value)

The answer given is "is less than 1/(n+1)." How to do this? Please explain
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1answer
148 views

Applying Watson's lemma $\int^{\infty}_{0}\{1+\sin(t^2)\}e^{-xt}dt$

So firstly Watson's lemma states that for $\phi(t)=t^\lambda g(t)$, where $g(t)$ has Taylor series $g(t)=\sum^{\infty}_{n=0}\frac{t^n}{n!}\frac{d^ng}{dt^n}(0)$ about $t=0$, with $g(0)\neq 0$, and $\...
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1answer
50 views

Is it integration or not

I was solving a physics problem and I encountered to find the sum the function $xe^{-kx}$ where for all positive values of $x$ (i.e. from 0 to infinity) Is there any way to do that? I speculate that ...
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1answer
50 views

Integral involving binomial expression of an exponential

I am trying to understand the behavior of the following function w.r.t $b$: $$ \mathrm{M}\left(b,k\right) = \int_{0}^{\infty}\mathrm{e}^{-kt}\left(2\mathrm{e}^{t} - 1\right)^{b} \,\mathrm{d}t\quad \...
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1answer
25 views

Quadrature rules estimation

Given is $$ I(f) = w_0 f(-1) +w_1 f(0)+w_2 f(1) + w_3 f(2) $$ I need to calculate the $w_i$ so its exact value of $\int _0^1 f dx$ for polynoms with grade 3. Now i want to show that $|\int _0^1 f dx-...
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0answers
39 views

Integration of a function involving algebraic and trigonometric functions [duplicate]

Evaluate $$f(x) = \int_0^{\pi/2}\frac{1}{(1+x^2)(1+\tan{x})}dx$$ My attempt: I could not apply any standard method known to me to solve this integration. The only way I thought of is expressing $\tan(...
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1answer
38 views

$\partial_\rho\bigg (\int_0^{\infty} \frac{\rho_0^2 J_1(q \rho_0)}{(a^2+(\rho_0-\rho)^2)^{\frac{5}{2}}} d\rho_0 \bigg)$

Is there any method to calculate above integral? I need to know answer for different $q$'s. $J_1$ is Bessel function. $\partial_\rho\bigg (\int_0^{\infty} \frac{\rho_0^2 J_1(q \rho_0)}{(a^2+(\rho_0-\...
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0answers
97 views

Name of midpoint rule and trapezoidal rule in higher dimension

Consider the integral $$ I = \int_0^a \int_0^b f(x,y)\,dy\,dx. $$ I can, in order to approximate the integral, use some kind of "trapezoidal rule" $$ I \approx I_{\mathrm{tr}} = \frac{ab}4\bigl(f(0,0)...
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0answers
94 views

Control speed of convergence of Riemann sum of Gaussian function

Hi this is my first question so far so I hope I'm doing it the right way. I'm trying to prove some result regarding the speed of congvergence of the Riemann sum $\Phi(R,\delta):=\sum_{k \in \mathbb{Z}...
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1answer
45 views

Numerical Integration of a singular function

I have a following function: $$f(r)=\int_{r}^{\infty}\frac{g(y)\mathrm{d}y}{\sqrt{y^2-r^2}}~,$$ where $g(y)$ is a continuous function and $g(r)\neq 0$ and $g(r\rightarrow\infty)=0$. Now, in reality I ...
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0answers
89 views

Combining Quintic Spline Interpolation with Mollifier Functions

I am using quintic splines for creating smooth contours especially around the edges. Although it gives me continuous third derivatives the values are too big. Would it be possible to limit it by ...
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1answer
70 views

Asymptotic approximation of an integral.

How to find the asymtotic approximation of this Fourier integral : $$I[T]=\lim_{T->\infty}^{}\frac{T}{2\pi}\int_{-\pi/T}^{\pi/T}d\lambda e^{iT\lambda}e^{Tf(\lambda)}.$$ $\textbf{Context}$ : This is ...
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1answer
59 views

Using the midpoint rule

QUestion: needed help here. i had tried it out but i get an answer that is not in the question answer list. i get 0.846474251 and i just assumed 0.63689453. see my workings below i used a tabular ...
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0answers
79 views

Integrate over a 512 dimensional cube

Basically, I'm trying to find what the mean and standard deviation of the distance between two random "hyperpoints" from a 512 dimensional hypercube of size 2 would be. I was thinking the mean would ...
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1answer
42 views

Cannot Show Bound of Integral Involving Strange Integration Region

$$(1)=\int_{n-1}^{\infty}\left(\int_{1}^{\infty}\cdots\int_{1}^{\infty}(v_1\cdots v_{n-1})^{-\beta-1}[v_1+\cdots+v_{n-1}\geq u] \;dv_1\ldots dv_{n-1}\right)du,$$ for any integer $n>1$ and a fixed $...
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0answers
45 views

Approaching an integral with the Simpson type

Lets say $f(x) = e^x$ and we approach the integral $$\int_a^b f(x)\, dx$$ with the Simpson type $$Q(f)=(b-a)/6{f(a)+4f((a+b)/2)+f(b)}$$ how can i prove that: $\int_a^b f(x)\,dx < Q(f)$? (also $...
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0answers
110 views

Calculate Error term for Integration interpolation

Let $x_0=a,x_n=b,x_i=x_{i-1}+h$ for $i=1,\cdots,n-1$ with $h=\frac{b-a}{n}$ and consider the (n+1)-points Newton-Cotes formula: \begin{equation}\sum_{i=0}^na_if(x_i), \ \ a_i=\int_{x_0}^{x_n} \...
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2answers
280 views

Taylor series and approximations to complicated integrals

So, I feel like I should technically know how to do this, but I'm really not sure. I have a certain integral to calculate or rather, approximate since it's quite hard (if you'd like to know, it's the ...
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0answers
39 views

Is there an approximation function for this function?

The function is: $$f(x,y)=\frac{x}{\log\left(1+\frac{1}{1+y}\right)}$$ where $x\in [0,1], y \in [0,1] $,and the approximation function can split $x$ and $y$. The result is like this:$f(x,y) = f_1(x)+...
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1answer
95 views

Trapezoidal rule

The trapezoidal rule is given by the following formula: $\int_{a}^{b}f(x) \approx \frac{b-a}{2}[f(a)+2\sum_{i=1}^{n-1}f(x_i)+f(b)] $ I have been given a question which wants me to evaluate the ...
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2answers
84 views

Approximating $\int \frac{1}{1 + a p^4 + b p^6}$

I'm attempting to calculate an approximate "closed form" of the integral $$\int \frac{dp}{1 + a p^4 + b p^6}$$ as a function of $a$ and $b$, two small parameters (of the order of $10^{-2}$). I'm ...
4
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0answers
55 views

Approximating an integral as a parameter grows large

I am trying to calculate the following integral : $$I(\lambda,\alpha)=\int_{\lambda}^1 \mathrm{d}\tau \frac{1-\tau^\alpha}{1-\tau}\exp(-k \tau)$$ where $\lambda<1$, $k$ is a positive constant ...