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Results tagged with closed-form-expressions
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user 13972
For questions that specifically ask for determining a closed form of equations, integrals etc.
162
votes
Accepted
Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?
The answer is 'no'. Making the substitution
$$
x = \frac{(t-1)(t-5)(t^2+2t+5)}{16t^2},
$$
one finds
$$
{\textstyle\sqrt{x+\sqrt{x+\sqrt{x+1}}}\,\mathrm{d}x}
= \frac{(t^2-2t+5)(t^2-5)\sqrt{t^4{-}2t^2 …
- 108k
108
votes
Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?
I'm adding a separate answer for the general question that the OP asked, which settles the question in the negative for all $n>2$ (and gives an alternate proof for $n=3$ to the one I gave above).
Rec …
- 108k
8
votes
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$
There always exists at least one solution: If $x >\mu_i$ for all $i$, then each of the terms in the sum is positive and if $x < \mu_i$ for all $i$, then each of the terms in the sum is negative. Thus …
- 108k
6
votes
A second order nonlinear ODE
This ODE has some very interesting properties. If one clears fractions and writes it out as
$$
x(x+2y)(x-2y+1)\,y'' = (4x^2-8y^2+3x+4y)\,y' + x(4y-1)\,(y')^2,
\tag1
$$
one recognizes this as the equa …
- 108k