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Tagged with nested-radicals definite-integrals
13
questions
17
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2
answers
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Crazy integral with nested radicals and inverse sines
Recently a friend who is writing a book on integrals added this problem to his book:
$$\int_{0}^{1}\arcsin{\sqrt{1-\sqrt{x}}}\ dx=\frac{3\pi}{16}$$
After a while, when trying to generalize, I was able ...
- 105
36
votes
1
answer
3k
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Nested root integral $\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$
The bigger goal is to find the antiderivative:
$$\int \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}~~~~~(*)$$
But I can settle for the definite integral in $(0,1)$. Motivation:
$$\int \frac{dx}{\sqrt{x+\...
- 28.4k
3
votes
0
answers
64
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Does this ridiculous integral converge?
I was looking through some of my older questions, when I came across this crazy integral I posted.
$\int\limits^{2}_{1}\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}dx$
I had approximated it at ...
- 1,688
1
vote
0
answers
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A very interesting integral: $\int\limits^{2}_{1}\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}dx$
I really like infinitely nested radicals so when I was messing around with them in Desmos, I noticed that the domain of $\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}$ seemed to converge to $1&...
- 1,688
10
votes
1
answer
277
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Evaluate: $\int_0^1 \sqrt{x+\sqrt{x^2+\sqrt{x^3+\cdots}}}\, dx. $
Is there a way to evaluate the integral:$$\int_0^1\sqrt{x+\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+\cdots}}}}\,dx,$$ without using numerical methods?
The integrand doesn't seem to converge to anything for any ...
- 4,472
0
votes
0
answers
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How to evaluate the integral $\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\cdots\sqrt{1+x}}}}}dx=?$
We have :
$$\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+x}}}dx=\frac{8}{315}\sqrt{2}\Big(16+\sqrt{233+317\sqrt{2}}\Big)$$
We are lucky because this integral have an anti-derivative like here.
More ...
15
votes
1
answer
417
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An incorrect answer for an integral
As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed).
$$
''\int_{0}^{\infty}\frac{1}{\left(1+...
- 13.8k
14
votes
1
answer
414
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Unsure of my work evaluating $\int \frac{dx}{\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}}$
This Question is an Extension of this Previously Asked Question: Nested root integral $\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$
I was looking into answering the question of whether it was ...
- 7,730
8
votes
2
answers
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Solve $ \int_0^2 \sqrt{x+\sqrt{x+\sqrt{x+\dotsb}}}\,dx $
I haven't seen this question, but if someone has, it would be very appreciated if you could send a link!
I've been very interested in the MIT Integration Bee, and one question that stood out to me ...
- 25.2k
12
votes
1
answer
386
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How would you prove $\int^8_0\frac1{\sqrt{x+\frac1{\sqrt{x}}}}dx<4-\frac1{2019}$?
We would like to prove the following inequality.
$$\int^{8}_{0}\frac{1}{\sqrt{x+\frac{1}{\sqrt{x}}}}\,dx<4-\frac{1}{2019}\tag{1}$$
What I've tried is using the AM-GM inequality, $$x+\frac{1}{\...
- 131
0
votes
1
answer
99
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An approximation for an integral involving nested radicals and logarithms
Yesterday I wrote some simple integrals involving nested radicals, and/or continued fractions. This was an example with nested radicals, let $$\int_0^1\frac{dx}{\sqrt{1+(\log x)g(x)}},\tag{1}$$ where $...
- 268k
26
votes
2
answers
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How to find $\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n$
Here I mean the limit of the following sequence:
$$p_1=\int_0^1 \sqrt{x} ~dx=\frac{2}{3}$$
$$p_2=\int_0^1 \int_0^1 \sqrt{x+\sqrt{y}} ~dxdy=\frac{8}{35}(4 \sqrt{2}-1) = 1.06442\dots$$
$$p_3=\int_0^...
- 59
2
votes
1
answer
133
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On the integral $\int_0^1 \frac{dx}{\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+\cdots}}}}$ and the plastic constant
We have,
$$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
$$P=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\cdots}}}$$
with golden ratio $\phi$ and plastic constant $P$. If,
$$\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\...
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