All Questions
Tagged with closed-form calculus
835
questions
3
votes
0
answers
91
views
What is $ \int_{0}^{\exp(-1)} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx $?
Background
According to p. 22 of the following paper by Blagouchine, we have the following Malmsten integral evaluation: $$ \int_{0}^{1} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx = \frac{\pi}{2} \ln\left(...
- 7,148
1
vote
3
answers
119
views
Is there any nonlinear function $f(x)$ for which both $\sin (f(x))$ and $\cos(f(x))$ have closed-form integrals?
I'm interested in closed-form antiderivaties of elementary functions.
The question arises from seeing that something as simple as $\int\sin(x^2)dx$ and $\int\cos(x^2)dx$ generates the Fresnel ...
1
vote
2
answers
148
views
Is there a closed form for the integral $\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$, where $k$ is a natural number?
To start, I am aware that our integral $I(k)=\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$ is equal to $$I(k)=\int_{0}^{\pi/4}x^{k}\ln\sin x\, dx-\int_{0}^{\pi/4}x^{k}\ln\cos x\, dx$$, but I cannot seem to ...
- 586
0
votes
2
answers
140
views
How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{4y \, da \, dy}{(y^2 + ay + 1)(y^2 - ay + 1)}$? [duplicate]
Question: How to evaluate $$\int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \sin x} \right) \, dx$$
My attempt
The original integral is:
$$
J = \int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \...
8
votes
1
answer
235
views
How to solve $\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}dx$
One of my friends sent me a list of integrals (all without solutions ) one of those problems is: $$\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}...
- 6,620
8
votes
3
answers
264
views
How to evaluate $\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx$?
Question: How to evaluate $$\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx?$$
My attempt
We will denote the main integral as $\Omega$.
$$\Omega=\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, ...
5
votes
2
answers
239
views
How can I show that $\int_0^{\frac{\pi}{2}}\sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx$.
Question: How can I show that
\begin{align}
& \int_0^{\frac{\pi}{2}} \sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right)\,dx
\\[2mm] = & \
{\small\log\left(\left(2\sqrt{2-\sqrt{2}}...
7
votes
2
answers
114
views
How to Evaluate the Integral $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$
Question: How to Evaluate the Integral $$\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$$
My attempt
I'm looking for a method to evaluate it. I've attempted a substitution to ...
2
votes
2
answers
138
views
A question about sum with reciprocal quartic
Evaluate
$$
\sum_{n=1}^{\infty} \frac{n+8}{n^{4}+4}
$$
According to WolframAlpha it is $\pi\coth(\pi) - \dfrac{5}{8}$
My attempt:
I tried to separate $\dfrac{n}{n^{4}+4}$ and $\dfrac{8}{n^{4}+4}$. ...
- 192
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
8
votes
4
answers
426
views
how to evaluate $\int_0^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}$
Integral: how to evaluate $$\int_0^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}$$
Same context
I'm not sure of the closed form of the integral, as I haven't evaluated it yet. ...
1
vote
0
answers
81
views
Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $ [closed]
After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum:
$ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\...
- 923
7
votes
1
answer
171
views
Least number of circles required to cover a continuous function on a closed interval.
Now asked on MO here.
This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to ...
- 6,620
7
votes
3
answers
431
views
How to evaluate $\int_0^1 \ln ^3(1+x) \ln (1-x) d x$?
QUESTION:How to evaluate $$\int_0^1 \ln ^3(1+x) \ln (1-x) d x$$?
I'm not sure of the closed form of the integral, as I haven't evaluated it yet. However, after evaluating the integral $$\int_0^1 \ln (...
1
vote
0
answers
66
views
How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$
here is my attempt to solve the integral
\begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
9
votes
1
answer
384
views
evaluation of $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n} H_{n+1}^{(2)}}{(n+1)^{2}}$ and other Euler sums
I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx $$
Here is my attempt so solve the integral
\begin{align}
&\int_{0}^{1} \frac{\ln (x) \ln^{...
user1313975
7
votes
1
answer
295
views
how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$
Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$
MY try to evaluate the integral
$$
\begin{aligned}
& I=\int_0^{\infty} \frac{x \...
user1313975
7
votes
3
answers
228
views
Closed form for $\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$?
Is there a closed form for $I=\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$ ?
Context
Earlier I asked "Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\...
- 25.7k
2
votes
1
answer
78
views
How to evaluate $\int_1^{\infty}\frac{t^2\ln^2 t\ln(t^2-1)}{1+t^6}{\rm d}t $
I was evaluating Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm
d}x .$
On the path of integrating the main function, I am stuck at this integral. I don't know how ...
4
votes
3
answers
220
views
Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$
Problem: Show that $$\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$$
Some thinking before trying
At least we ...
12
votes
1
answer
653
views
Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.
There is numerical evidence that
$$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$
How can this be proved?
Context
In another question, three random ...
- 25.7k
4
votes
2
answers
238
views
Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $
Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
- 6,620
20
votes
1
answer
1k
views
Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.
There is numerical evidence that
$$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$
How can this be proved?
...
- 25.7k
2
votes
0
answers
247
views
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 ...
6
votes
0
answers
104
views
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,...
- 1,632
14
votes
1
answer
637
views
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 ...
- 25.7k
0
votes
1
answer
120
views
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
$$\int_{2}^{\infty} \frac{\pi x (1 + x^2) \log^2(x^{\...
3
votes
1
answer
241
views
Remarkable logarithmic integral $\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} dx$
Question: how to evaluate this logarithm integral?
$$
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
votes
1
answer
183
views
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, ...
- 399
2
votes
2
answers
148
views
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
$$\int_{0}^{1} \int_{...
1
vote
1
answer
152
views
How to evaluate $\sum_{n=1}^{\infty}\frac{H_{n}^{(2)}}{n^{4}}$
It may be rather tedious and I will have to delve into deeper, but I have a little something. We probably already know this one. The thing is, the first one results in yet another Euler sum. But, I ...
7
votes
2
answers
285
views
Show that $\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}$
Show that\begin{align*}
\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}
\end{align*}where $a\in \mathbb{R}$.
My SOLUTION
Let $\...
0
votes
0
answers
132
views
Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
3
votes
1
answer
180
views
How to integrate $\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$
Question
How to integrate $$\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$$
My attempt
\begin{align*}I &= \int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^...
0
votes
2
answers
110
views
Show that $\int_0^\infty \frac{x^2+1}{x^4+x^2+1}\frac{\log(1-x+x^2)}{\log(x)}dx=\frac{\pi}{\sqrt{3}}$
$$\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}}...
4
votes
1
answer
140
views
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.
...
6
votes
0
answers
172
views
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 $$\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx=\dfrac{\pi^...
2
votes
0
answers
81
views
how to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$
how to evaluate $$\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}...
14
votes
4
answers
665
views
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
$$ \int_{0}^{\frac{\pi}{2}} \frac{\cos(...
4
votes
2
answers
200
views
How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$
How to evaluate this sum $$\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} \...
4
votes
2
answers
260
views
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
$$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,...
4
votes
1
answer
171
views
How to integrate $\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$
How to integrate
$$\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\...
1
vote
0
answers
188
views
Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ ...
- 16.4k
2
votes
1
answer
115
views
How to integrate $\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx$
How to integrate $$\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx$$
Let $z = e^{ix}$. We write the Fourier series
$$
\frac1{\tan^2x+\cot^2x} = \frac1{\left(\frac{z-z^{-1}}{i(z+z^{-1})}\...
6
votes
2
answers
165
views
Integrating $\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$
how to integrate $$\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$$
Attempt
$$=\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx = \int_{0}^{1} \frac{1}{x^4} \cdot (\arctan(x) -...
2
votes
1
answer
99
views
Integrating $\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x$
Show that
$$\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x=\boxed{2\log\left(\frac{768\sqrt2\pi^4}{25\Gamma^3\left(\...
4
votes
2
answers
211
views
How to calculate this integral $\int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$
How to calculate this integral
$$ \int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$$
This is how I start
$$f(a)=\int_{0}^{\infty} \frac{(x^a-1)^2}{\ln^2(x)}\frac{1}{...
5
votes
3
answers
461
views
Integration of $ \int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx $ [closed]
Question: What is the closed form of this following integral? $
\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx.$
Here is my solution
we know that
$$\displaystyle{\sum\limits_{n = 1}^\infty {\frac{{\...
2
votes
2
answers
227
views
Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$
Question statement
Evaluate the infinite product
$$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$
My try
Because of the square of $\displaystyle{x}$ , we can consider $...
5
votes
5
answers
376
views
How to evaluate $\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$
Question
$$\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$$
Wolfram alpha says it is
$$\int_{-\infty}^{\infty} \frac{\cos(x)}{\left(1 + x + x^2\right)^2 + 1} \,dx = \frac{\...