# Tagged Questions

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### How large should $a$ be so that $\int_a^{\infty} \frac{dx}{1+x^2} < \frac{1}{1000}$

I want to solve this without using calculator.
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### integral approximation (law of large numbers)

I am totally at a loss with this question and don't even know where to begin. Let $g:[0, 1]\rightarrow \mathbb{R}$ be a measurable and Lebesgue-integrable function. $U_1, U_2, \dots$ be a series of ...
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### Why is Simpson's rule exact for cubics?

I can't understand: Why is Simpson's rule exact for cubic polynomials?
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### Show $\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$?

is there a way to show the inequality $$\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$$ for positive constants $M$ and $C$ ...
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### Numerically integrating in to Chebyshev polynomial

I'm trying to find the Chebyshev interpolate for an ODE in a given interval. That is, given an ODE that looks something like: $$y'' = g(y) \ y'$$ I want to numerically integrate it inside the ...
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### Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
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### Taylor series of an integral

I have the following integral $$2\int_r^\infty \frac{x g(x)}{\sqrt{x^2-r^2}}\text{d}x$$ where $g$ is a probability distribution (normalized and symmetrical around its only maximum in 0). I'm ...
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### Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$

I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
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### Analytical approximation of an integral

I think there is no analytical solution for $$\int_{K}^{\infty} \frac{exp(-x)}{x} dx$$ where $K > 0$. Instead, is there an analytical approximation?
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### Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?

Is there an asymptotic expansion for the function: $$g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du,$$ over the domain $x\in [0,\infty)$ in terms of ...
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### Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
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### Laplace's method with unknown exponent.

Given the integral: $$I = \int_0^a{e^{-\lambda g(x)}f(x)dx}$$ Where $g(x)$ and $f(x$) are both low order positive polynomials, and $\lambda \gg 1$, Laplace's method is commonly used to approximate ...
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### Remainder term for Gauss-Laguerre quadrature

I need to calculate a quadrature rule with maximum degree of accuracy that looks like this: $$\int_0^\infty e^{-x}f(x)dx = \sum_{i=0}^n A_if(x_i) + R_n(f)$$ where $n=2$. For $R_n(f)$ I have this ...