# Tagged Questions

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### Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
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### Correlation 4-point

I need to calculate $\langle x_{i}x_{j}x_{k}x_{l}\rangle$, where $$\langle f(x) \rangle = \int e^{-\frac{1}{2}A_{ij}x^{i}x^{j} - \frac{\lambda }{4!}\sum_{i}x_{i}^{4}} f(x)d^{n}\mathbf x ,$$ for ...
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### Copying the Curvature of One Function onto Another: Approximation

I have a polar function $$r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)}$$ Is it possible to methodically conjure another polar function ...
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### Help evaluating or approximating this integral

For a thermodynamics project I'm working on, I need to evaluate this integral: $\int \frac{(a-bx)(x-c)^d}{x^3}dx$, where $a,b,c,$ and $d$ are all positive constants. I tried evaluating it on ...
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### How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm trying to use ...
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### Approximate the integral $\int_0^\pi \sin(x^3)\mathrm{d}x$ with a standard pocket calculator

I came over the following integral $$\int_0^\pi \sin(x^3) \mathrm{d}x$$ when a friend of mine tried to approximate it. The most obvious way is to use taylors formula, and then turn the integral ...
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### alternating series estimation with integral?

We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C})$$ Then if $X\ge\lambda_1$ and $\phi(x)$ has ...
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### Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
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### Integral approximation.

Can you help me to show that $$\int^{\pi/2}_{0}{d\theta\over (1-m^2\cos^2\theta)^2} \approx {(2-m^2)\pi\over4(1-m^2)^{3/2}}$$ to first order, such that $0 \lt m \lt 1$
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### approximate error between integral an sum

I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by ...
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### integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
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### Estimate the area restricted by $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$.

I need to estimate the area between the functions $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$. where $a>1$. Now I have tried quite a few ways to do this, but nothing comes to mind. I ...
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### asymptotic limit of $\int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt$

Help me please with the following integral. I've asked this question before Asymptotic limit of the integral with polynomial, but it turns out that it was incorrect question. I should get an ...
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### Algorithms for finding closed form approximations for integrals (with no closed form solutions)

It is well known that many integrals have no closed form solutions, normally what you would do is solve them numerically. My question is if there are algorithms that give you good closed form ...
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### Approximation of Integration by parts

I'm trying to approximate a integral of the form: $$\int_V{g({\bf x})f({\bf x})} \; d^3x$$ Where the functions $f(\bf{x})$ and $g(\bf{x})$ are positive functions, but where only $f(\bf{x})$ is known ...
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### Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
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### How to approximate an integral using the Composite Trapezoid Rule

I'm trying to estimate the value of the following integral on the interval $[0,1]$ $$I = \int_0^1 \frac{1}{1+x} dx$$ So, using the composite trapezoid rule (and with $n=4$, ie I'm only using the ...
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### Projection of Gaussian in Spherical Coordinates

Consider a point with spherical coordinates $\vec{r}_0=(r_0, \theta_0, 0)$. The spherical gaussian distribution centered at $\vec{r}_0$ is $f(\vec{r})=Ne^{|\vec{r}-\vec{r}_0|^2/A}$, where $N$ is the ...
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### Approximate $\int_{0}^{\infty} \frac{\text{d} x}{1 + x^4}$

Now, I have been given this integral. And need to approximate it. My first idea was to use a Taylor series, but this series explodes, as x reaches infinity. Does anyone know how to approximate ...
I'd like to approximate a sum of the form $S(n)=\sum\limits_{k=1}^{n}\phi\left(\frac{k}{n}\right)$ with an integral using Riemannian sums: $$S(n) \approx n \int_{0}^{1}\phi(x)dx +o(n).$$ My ...