All Questions
Tagged with closed-form calculus
835
questions
3
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0
answers
87
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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: $$ \int_{0}^{1} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx = \frac{\pi}{2} \ln\left(...
- 2,638
7
votes
1
answer
171
views
Least number of circles required to cover a continuous function on a closed interval.
Now asked on MO here.
This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to ...
- 6,620
21
votes
4
answers
2k
views
Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$
Does the integral below have a closed-form:
$$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$
where $\tan^{-1} (\cdot)$ is inverse tangent function.
...
5
votes
3
answers
199
views
Closed form: $\displaystyle\int_0^\infty \! \prod_{i=1}^n\frac{1}{a_i^2 + x^2} \, dx$
Can you find the closed form for the following integral
$$ \int_0^\infty \prod_{i=1}^n \frac{1}{a_i^2 + x^2} \, dx = \int_0^\infty \! \frac{dx}{(a_1^2 + x^2) (a_2^2 + x^2) \cdots (a_n^2 + x^2)} $$
...
- 380
8
votes
3
answers
264
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How to evaluate $\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx$?
Question: How to evaluate $$\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)) \, ...
- 91k
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}}...
- 91k
1
vote
3
answers
119
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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 ...
- 7,212
37
votes
4
answers
2k
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Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
Find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}.$$
We can use the Fourier series of $e^{-bx}$ ($|x|<\pi$) to find $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}.$$ But here ...
- 2,308
8
votes
4
answers
426
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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. ...
- 28.4k
1
vote
2
answers
147
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 ...
- 99.7k
8
votes
1
answer
235
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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: $$\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}...
- 6,620
0
votes
2
answers
140
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How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{4y \, da \, dy}{(y^2 + ay + 1)(y^2 - ay + 1)}$? [duplicate]
Question: How to evaluate $$\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 - \...
- 268k
5
votes
3
answers
461
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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
$$\displaystyle{\sum\limits_{n = 1}^\infty {\frac{{\...
5
votes
1
answer
125
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A challenging Integral Involving Logarithmic and Trigonometric Functions
Question: How to evaluate $$\frac{1}{2\sqrt{2}} \int_{0}^{\frac{\pi}{2}} \frac{\log(1 + \tan y)}{(\cos y + \sqrt{2} \sin(y + \frac{\pi}{4})) \sqrt{1 + \sqrt{2} \sin(2y + \frac{\pi}{4})}} \, dy = G$$
...
6
votes
0
answers
172
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How to evaluate $\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
crossposted: https://mathoverflow.net/q/464839
How to evaluate $$\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^...