Linked Questions

12
votes
5answers
996 views

Why has $\int \sin (\sin x) dx$ not been solved yet?

I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation. What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what ...
40
votes
1answer
3k views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ Where $...
32
votes
1answer
695 views

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

I encountered this scary integral $$\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$$ where $_3F_2$ is a generalized hypergeometric function $$_3F_2\...
28
votes
2answers
1k views

Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit means ...
4
votes
3answers
3k views

Solve equation $\exp(ax)+\exp(bx)=1$

The equation is $$ \exp\left(ax\right)+\exp\left(bx\right)=1, $$ where $a$ and $b$ are known real constants, $x$ is unknown. I would like to have the solution in form of relatively known special ...
1
vote
1answer
143 views

Do elliptic allow for direct solvers of roots of quintic polynomials?

Galois Theory tells us that we cannot directly solve for the roots of a quintic polynomial using elementary operations and radicals. I have seen sources that use this to reason that any computer ...
2
votes
0answers
234 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
0
votes
1answer
74 views

Explicit solution of the following equation possible?

Is it possible to obtain an explicit solution for $K$ for the following equation? $$(e^K - 1)(e^{\beta K} - 1) = q$$ for $0\leq q \leq 4$ and $0\leq \beta \leq 1$ For $\beta=1$ one gets $K=\log{\...
1
vote
0answers
79 views

Is it possible to find solutions to polynomials purely by calculus and without iteration?

I know this may sound peculiar, but I was wondering if any mathematicians have found a method to finding roots purely through calculus without iteration. I can't imagine that such a method exists for ...
3
votes
0answers
60 views

Is there a formula which gives the solutions of a general fifth grade equation by using trascendental functions? [duplicate]

Thanks to Galois we know that the solutions of a general fifth order equations cannot be expressed by means of radicals. Can we exclude that a formula exists if we are allowed to also use (any ...