Questions tagged [closed-form]
A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
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Is it possible to evaluate $\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$?
How to evaluate $$\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$$
Source: I created this integral so I don’t know the closed form
I tried Wolfram Alpha, but Wolfram Alpha is unable to ...
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Closed form of $\int_{x}^{1}\left(1-s\right)^{-1/2}P_{m}^{\left(1/2,-1/2\right)}\left(s\right)ds$
Let $P_{m}^{\left(a,b\right)}\left(s\right)$ the $m$-th Jacobi polinomial. An old result of Bateman shows that $$
\left(1-x\right)^{\alpha+\mu}\frac{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(x\...
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Why does the poisson regression beta coefficient ML estimate have no closed form solution?
Every source I have read says that there is no closed form solution for the beta coefficient but I have not seen an explanation as to why. I tried to solve for the beta coefficient on my own to see ...
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Please help me identify any errors in my solution to the following DE: $xf(x)-f'(x)=0$, $f(0)=1$
Context/background:
I am self-studying series, first in the context of generating functions and now in the context of functional/differential equations. As such, I like to set myself practise problems,...
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Show that $\int_{\arccos(1/4)}^{\pi/2}\arccos(\cos x (2\sin^2x+\sqrt{1+4\sin^4x})) \mathrm dx=\frac{\pi^2}{40}$
There is numerical evidence that $$I=\int_{\arccos(1/4)}^{\pi/2}\arccos\left(\cos x\left(2\sin^2x+\sqrt{1+4\sin^4x}\right)\right)\mathrm dx=\frac{\pi^2}{40}$$
How can this be proved?
I was trying to ...
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How to integrate $\int_{2}^{\infty} \frac{\pi(x) \ln(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx$
How to integrate $$\int_{2}^{\infty} \frac{\pi(x) \ln^2(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx \quad?$$
Wolfram gives the numerical value
$$\int_{2}^{\infty} \frac{\pi x (1 + x^2) \log^2(x^{\...
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Closed form of ${_2}{F}_1\left(\frac{1}{2},s,\frac{3}{2};-\frac{1}{a^2}\right)$
While solving an integral, I came acorss the term
$$ \tilde{I}(a,s)= {_2}{F}_1\left(\frac{1}{2},s,\frac{3}{2};-\frac{1}{a^2}\right).$$
To be precise, it came in the following calculations
\begin{align*...
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Remarkable logarithmic integral $\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} dx$
Question: how to evaluate this logarithm integral?
$$
I=\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x
$$
My attempt:
$$
\begin{aligned}
I=&\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x\\
...
3
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Improper Integral $\int_{0}^{\infty} \log(t) t^{-\frac{1}{2}} \exp\left\{-t\right\} dt$
Background
Hi. I am currently writing my undergraduate thesis which mainly revolves around the generalized log-Moyal distribution pioneered by Bhati and Ravi (see here). In the aforementioned article, ...
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Why is none trying to find the value of $\displaystyle\sum_{n=1}^\infty\frac1{n^n}$?
I wish to know the exact value of
$$\sum_{n=1}^\infty\frac1{n^n}.$$
I've found on the internet some mentions of the equality (the Sophomore's Dream, as I've learned)
$$\sum_{n=1}^\infty\frac1{n^n} = \...
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How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{{(1 + x) \cdot \log(x) - (1 + y) \cdot \log(y)}}{{x - y}} \cdot (1 + \log(xy)) \,dy \,dx$
Question: How to evaluate this integral $$\int_{0}^{1} \int_{0}^{1} \frac{{(1 + x) \cdot \log(x) - (1 + y) \cdot \log(y)}}{{x - y}} \cdot (1 + \log(xy)) \,dy \,dx$$
My messy try
$$\int_{0}^{1} \int_{...
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1
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How to evaluate $\sum_{n=1}^{\infty}\frac{H_{n}^{(2)}}{n^{4}}$
It may be rather tedious and I will have to delve into deeper, but I have a little something. We probably already know this one. The thing is, the first one results in yet another Euler sum. But, I ...
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Variants of geometric sum formula
I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$
and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$.
(See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From Sum of ...
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Show that $\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}$
Show that\begin{align*}
\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}
\end{align*}where $a\in \mathbb{R}$.
My SOLUTION
Let $\...
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Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
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How to integrate $\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$
Question
How to integrate $$\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$$
My attempt
\begin{align*}I &= \int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^...
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2
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Show that $\int_0^\infty \frac{x^2+1}{x^4+x^2+1}\frac{\log(1-x+x^2)}{\log(x)}dx=\frac{\pi}{\sqrt{3}}$
$$\displaystyle \int_0^\infty \frac{x^2+1}{x^4+x^2+1}\frac{\log(1-x+x^2)}{\log(x)}dx=\frac{\pi}{\sqrt{3}}$$
It involves Beta.
Start with $$\displaystyle \int_{0}^{\infty}\frac{x^{2n-2}}{(1+x^{2})^{2n}}...
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How to evaluate $\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx$
How to evaluate $$\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx = G - \gamma + \frac{1}{4} \pi \log 2 - \frac{3}{2}.$$
I made some progress.
...
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How to evaluate $\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
crossposted: https://mathoverflow.net/q/464839
How to evaluate $$\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx=\dfrac{\pi^...
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how to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$
how to evaluate $$\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$$
My attempt
$$ \Omega =\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}...
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What is the number of integers divisible by either 2, 3, or 5 from the integers 1 to $n+1$? [closed]
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
As joriki said, they are the number ...
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What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?
Let all untyped variables be natural numbers.
Formula?
Given $x \geq 1$, $0 \leq r \lt d$ there are two cases to handle: $x \lt r$ and $x \geq r$.
What is the 2-case closed form for $\sum_{i = 1}^{x} \...
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How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$
How to evaluate $$\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$$
My attempt
The transformation of $x \rightarrow \frac{\pi}{2}-x$ yields
$$ \int_{0}^{\frac{\pi}{2}} \frac{\cos(...
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approximate a sum [closed]
Is there a way to simplify the given function:
$f(n):={\sum}_{x={\lceil\frac{n}{\mathrm e}\rceil}}^{n}\frac{\ln\left(\frac{n}{x+1}\right) \left(x+1\right)}{\left(x-1\right)
x}$, with $n>...
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How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$
How to evaluate this sum $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
My attempt
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
$$= \sum_{n=1}^{\infty} \...
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How to integrate $\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$
how to integrate $$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$$
My attempt
$$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,...
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Evaluate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x \text{d}y\text{d}z$
How to evaluate
$$
I=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}
\frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x
\text{d}y\text{d}z?
$$
Some simple calculation shows that
$$
I=\frac{\sqrt{2} -1}{4}\pi+\frac{\pi^...
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how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\...
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How to integrate $\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$
How to integrate
$$\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$$
My attempt
$$\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\...
5
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How to integrate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\frac{x^{4a - 1} \ln(x)}{\sqrt{yz} \cdot (1 + x^{2a}z + yzx^{4a} + yx^{2a})} \,dx \,dy\,dz$
how to integrate$$\int_{0}^{1} \int_{0}^{1}\int_0^1 \frac{x^{4a - 1} \ln(x)}{\sqrt{yz} \cdot (1 + x^{2a}z + yzx^{4a} + yx^{2a})} \,dx \,dy\,dz$$
My attempt
$x^{2a} \rightarrow x$
$$=\frac{1}{4a^2} \...