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Mathematics Stack Exchange

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.

Approximate integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

The basic problem in approximate integration is to compute an approximate solution to a definite integral $$ \int_a^b f(x)\;dx$$ to a given degree of accuracy. If $f$ is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired accuracy.

263 questions
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Solving an integral involving exponential functions $\int_{-\infty}^{+\infty} \frac{1}{\left(e^x+ e^{-x}\right)^n} e^{-\rho x^2 + a x} dx$

I need to calculate $$\int_{-\infty}^{+\infty} \frac{1}{\left(e^x+ e^{-x}\right)^n} e^{-\rho x^2 + a x} dx$$ where $n \in \mathbb{N}$, $\rho > 0$ and $a \in \mathbb{R}$, but I don't know how to follow. I've tried to include the expression in…
marc1s
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Dealing with an integral: can we go any farther?

I meet an integral, but it is beyond my ability. $$ {\rm I}\left(a\right) = \int_{a}^{1}{\arcsin\left(\,\sqrt{\,{1 - x^{2} \over 1 - a^{2}}\,}\,\right) \over x + 1}\,{\rm d}x, 0\le a <1. $$ I can work it out when $a = 0$, but failed otherwise. I…
Roger209
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How to obtain asymptotic form of the integral $\int_0^\infty \frac{dx}{x^2 + 1} e^{-\alpha \sqrt{x^2 + 1}}$ for small and large values of $\alpha$?

I am trying to evaluate the following integral as a function of $\alpha$: $$ \operatorname{I}\left(\alpha\right) = \int_{0}^{\infty}\frac{{\rm d}x}{x^{2} + 1} {\rm e}^{-\alpha\sqrt{\,x^{2} + 1\,}} $$ $\operatorname{I}\left(0\right) = \pi/2$ and for…
Archisman Panigrahi
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Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$

This is my first question here, so I hope I'm not giving too little/too much information. I need some help calculating (or even approximating) an integral which I've been wrestling with for a while. As part of my internship, I need to calculate or…
Philip
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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 integration region as the form of a…
NN2
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Riemann Sum Approximations: When are trapezoids more accurate than the middle sum?

We can approximate a definite integral, $\int_a^b f(x)dx$, using a variety of Riemann sums. If $T_n$ and $M_n$ are the nth sums using the trapezoid and midpoint (middle) sum methods and if the second derivative of $f$ is bounded on $[a, b]$ then…
futurebird
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Analytical approximation of integral of Bessel function

I am trying to approximate the integral: $$ \int_0^z \left(\frac{J_1(x\,\sin\theta)}{\sin\theta}\right)^2 {\rm d}\theta $$ My very naive approach was to do the Taylor series of the integrand. However, for rather large x, I need to have increasingly…
atapaka
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Integrate a weighted Bessel function over the unit disk

I would like to evaluate a complex-valued integral of the form $$ I_e = \int_0^1 x e^{iax} J_0(b \sqrt{1-x^2}) dx $$ where $a$ and $b$ are real numbers (not necessarily positive) and $J_0(z)$ is the Bessel function of the first kind. An alternative…
user778563
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What is the position of the maximal value of this bell-shaped function?

Consider the following function : $$\tag{1} f(v) = v^{\frac{d}{2}} \int_v^{\infty} u^{\alpha \,-\, \smash{\frac{d}{2}} \,-\, 1} \; e^{-\, \alpha \, u} \; du, $$ where $d \le 6$ and $\alpha > 0$ are two positive constants (parameters). Notice the…
Cham
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Approximate solution of differential equation

My task: find approximate solution as $$y = y_0(x) + y_1(x)\lambda + y_2(x)\lambda^2 + y_3(x)\lambda^3$$ of differential equation $$y' = \sin x + \lambda e^y, y(0)=1-\lambda. \ \ \ \ (*)$$ My attempt : Let $$y(x,\lambda) = y_0(x) +…
Simankov
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What is the value of $\lfloor \int_{0}^{2023}\frac{2}{x+e^x}dx \rfloor$?

What is the value of $\lfloor \int_{0}^{2023}\frac{2}{x+e^x}dx \rfloor$? My attempt: If $t$ is smaller than $2023$ and large enough, then $\int_{t}^{2023}\frac{2}{x+e^x}dx\approx\int_{t}^{2023}\frac{2}{e^x}dx=2(e^{-t}-e^{-2023})\approx 2e^{-t}$. I…
佐武五郎
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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 one in this question and it's been quite painful.…
Philip
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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 and $\alpha$ is a large integer. I was thinking of…
Someone1348
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How to approximate the integral $\int_{-b}^{\infty}\log(t+b)e^{-t}e^{-e^{-t}}dt$

Suppose we have the following integral \begin{equation} \int_{-b}^{\infty}\log(t+b)e^{-t}e^{-e^{-t}}dt, \end{equation} where $b$ is a positive constant. It seems very difficult to derive the exact result. So my question is: is there any…
kawofengche
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Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where $g(x_0)$ is the maximum of $g$ and $g''(x_0)$ is…
alfC
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