# Tagged Questions

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### How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
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### Change of variable

I have to approximate the following integral, using Simpson's Composite $1/3$ Rule: $\displaystyle \int\limits_{0}^1 \mathrm{\frac{e^{2x}}{\sqrt[5]{x^2}}}\,\mathrm{d}x$. The only problem is that ...
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### Comparison of trapezoidal , Simpson's 1/3 ,Simpson's 3/8 and Boole's rules.

These rules are often used in numerical integration. How do we analyze the given support points or function and select the most suitable one for best approximation?
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### How to Split a 2D Gaussian pdf into a Grid of Equally Sized Volumes

Let $f(x,y)$ be a Gaussian pdf for some known mean and covariance. Given $(x_0, y_0)$ and $(x_N, y_M)$ such that $$\int_{x_0}^{x_N} \int_{y_0}^{y_M} f(x,y) dy dx \approx 1$$ I would like to split ...
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### Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
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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 ...