# 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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### 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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### Saddle Point Approximation for Multiple Contour Integrals

Using the multinomial theorem $$\left(\sum_{j=1}^{N} x_j \right)^N = \sum_{m_1 + \cdots + m_n = N} \frac{N!}{m_1! \cdots m_N!} x^{m_1} \cdots x^{m_N},$$ and the contour integral expression of the ...
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### Arbitrarily large degree of exactness for a precise quadrature formula

I don't know how to solve the second part of the following exercise: Consider the quadrature formula $Q(f)=w_1 f(x_1)$ for the computation of the weighted integral $$W(f)=\int_0^1 x^{\alpha} f(x)dx$$ ...
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### Taylor expansion of a Lorentzian integral to second order

I am considering a Lorentzian integral of the following type: \begin{equation} \int_{-\infty}^\infty\mathrm dx\,\frac{\gamma^2}{(x-x_0)^2+\gamma^2}f(x), \end{equation} where $f(x)$ is a "nice&...
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### Monte Carlo Approximation of Integral Transform

I'm looking for bounds over the following Monte Carlo approximation. Define a subset $\Omega$ of R^n (compact, not sure if it is important) and probability density function $P(X)$ over it. Further, ...
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### Find values of $p$ such that trapezoidal rule gives second order approximation

I have the following question, which should be easy, but I'm puzzled now: Consider $f(x)=|x|^\frac{p}{2}$, where $p$ is a non-negative integer and the integral $\int_{-1}^{1}f(x)dx$. For which ...
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### changing variables for numerical integration

I want to numerically integrate a set of data in Python, but I think my question might a math than a Python question. where I have discrete data points L, f(L) and I need to perform this integral: ...
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### Newton-Cotes quadrature formula being “correct” for polynomials

So I'm going through a chapter on Newton-Cotes quadrature formulas and everything's clear up until a certain point. First let me just get the notation stuff out of the way so we can be on the same ...
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### Integral $\int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt$ [closed]

$$\int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt$$ Hello everyone,i would like to know the result of the above integral and how to calculate or estimate it. background and progress so far:(1)...
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### Error Estimation for Gauss Quadrature

I have an equation $$\int_{0}^{1}sin(x)dx$$ and I want to find the error estimation for Gaussian quadrature using the above equation. I saw this formula for finding the error of the gaussian ...
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### Error Estimate for f(x,y) using trapezoidal rule

Please I need help. How do I find the error estimate of the trapezoidal rule of the function $$\int_{0}^{1}\frac{x^{2}}{1+y^{3}}dy$$ using $$-\frac{h^{2}}{12}[f'(b)-f'(a)]$$ where $$x,y\in(0,1)$$ ...
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### Asking for help in evaluation of an integral in a research paper

I need Help in evaluating an integral. I have a function $$\rho_0(x) = \min_{ y\in\mathbb R} \rho( x,y) = \min_{0\leq y < 1} \rho(x, y),$$ which is $1$- periodic. $\rho_0(x)$ can be ...
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### Operation count for Trapezoidal Integration vs Romberg Integration

I have been working with computing and numerical integration, and I have to juxtapose the operations required for Trapezoidal and Romberg for $n$ points to a certain error tolerance. For trapezoidal ...
The maximum value of the integral $$\int_{a-1}^{a+1} \frac{1}{1+x^8}dx$$ is attained A) at exactly two values of $a$ B) only at one value of $a$, which is positive C) only at one value of $a$ ...