Recently Active Questions
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How to simplify the integral $\int \frac{1}{(1+x^2)^2}dx$? [duplicate]
If any integration is in form $$\int \frac{1}{1+x^2}dx$$ it easily follows by substituting for $\tan^{-1}x$. But how to simplify if we have
$$\int \frac{1}{(1+x^2)^2}dx$$
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Characterization of real numbers $ c $ for which $ \hat{\mu}(n) = e^{c i n^2}$ holds for a finite Borel measure $ \mu $ on $ \mathbb{T} $
Which real numbers $ c $ does the following hold for?
There exists a complex Borel measure $ \mu $ on the torus $ \mathbb{T} := \mathbb{R} / 2\pi \mathbb{Z} $ with finite total variation (i.e., $ |\mu|...
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What is the degree of the differential equarion $\sin\left(\frac{dy}{dx}\right)=x$?
Consider the differential equation $$\sin\left(\frac{dy}{dx}\right)=x$$ what is the degree of the above differential equation? According to me its degree is not defined as the equation is not ...
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Confusions about $\mathrm{Spec}(R)$ and algebraic groups
So I am studying about classical algebraic geometry (the one working with affine varieties instead of schemes) and I suppose I'm lost in a number of points. Note that I'm trying to study affine ...
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Identity involving the polylogarithm and $\pi i/ 6$
Using the polylog multiplication formula, $\sum_{m=0}^{p-1} \operatorname{Li}_s(z e^{2\pi i m/p}) = p^{1-s} \operatorname{Li}_s(z^p)$
it's easy to see that the the real part of
$- \operatorname{Li}_2(...
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Compute $\int_0^{\frac\pi2} \frac1{\sin(x+\frac\pi3)\sin(x+\frac\pi6)} dx$
Compute the following definite integral:
$$\int_0^{\frac\pi2} \frac1{\sin(x+\frac\pi3)\sin(x+\frac\pi6)} dx$$
I know that I can rewrite the denominator as shown below:
$$\int_0^{\frac\pi2} \frac1{\...
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What is the logic behind Householder decomposition?
I’ve been studying the Householder decomposition and have encountered a few conceptual questions I’m hoping to clarify:
Why are reflections used in this method instead of seemingly simpler operations ...
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2
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Is there a direct derivation to show that $\int_0^\infty \frac{1-x(2-\sqrt x)}{1-x^3}dx$ vanishes?
Came across
$$I=\int_0^\infty \frac{1-x(2-\sqrt x)}{1-x^3}dx$$
and broke the integrand as
$$\frac{1-x(2-\sqrt x)}{1-x^3}=\frac{1-x}{1-x^3}- \frac{x(1-\sqrt x)}{1-x^3}
$$
The first term simplifies and ...
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If $\int_a^b\ln\left(\frac{3+3x+x^2}{1+x+x^2}\right)dx=0$, find the value of $2|a+b|$
If $\int_a^b\ln\left(\frac{3+3x+x^2}{1+x+x^2}\right)dx=0$, find the value of $2|a+b|$. (Given that $a\ne b$)
My Attempt:
I added and subtracted $2x^2$ in the numerator but couldn't proceed from there....
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answers
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Integral domain without unity has prime characteristic?
By an integral domain, we mean here, a ring (not necessarily with unity) in which $ab=0$ implies $a=0$ or $b=0$.
Question: If an integral domain without unity has positive characteristic, is it ...
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Find $O(b)$ given that $a^5=e$ and $ab^{-1}a=b^2$
In a mathematics group chat, somebody asked this question.
If in a Group G, $a^5=e$ and $ab^{-1}a=b^2$ then find O(b)
I believe he made an error, as this problem is similar to Problem 37 on p.49 in ...
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Exponential Rate Analog of Berry-Esseen.
The Berry-Esseen theorem is as follows:
Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with mean $\mu$, variance $\sigma^2 > 0$, and third absolute moment $\rho = \mathbb{E}[|X_1 - \mu|^3] &...
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Characteristic of a Non-unital Integral Ring
If $R$ is a unital integral ring, then its characteristic is either $0$ or prime. If $R$ is a ring without unit, then the char of $R$ is defined to be the smallest positive integer $p$ s.t. $ pa = 0 $ ...
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The order 4 isometry of the tetrahedron
Consider a tetrahedron centered at the origin. For example with the vertices $ (\pm1,0,-1/\sqrt{2}),(0,\pm1,1/\sqrt{2})$. The isometry group of the tetrahedron is $ S_4 $ and the subgroup of ...
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Evaluate $I\left(n\right)=\int_{0}^{\infty}\frac{1}{\left(e^{x}+e^{-x}\right)^{n}}\,\mathrm dx$
Evaluate $I\left(n\right)=\int_{0}^{\infty}\frac{1}{\left(e^{x}+e^{-x}\right)^{n}}\,\mathrm dx$.
I tried substitution $t=e^{x}$, which led to the integral $$\int_{1}^{\infty}\frac{t^{n-1}}{\left(1+t^{...
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