What does one mean when $\int \frac{\sin x}x$ doesn't exist? Well I say that by taylor's expansion:
$$\int\frac{\sin x}x=\int\frac{x-x^3/6+x^5/120+...}x=x-x^3/18+x^5/480+...+\mathbb{C}$$
It's another thing that there doesn't exists a closed form for the sum/difference.But it does exists.So I am now confused about:


*

*Does integration to every function exists?

*

*[I partly understand something told about elementary functions etc.]


*If it does, does there exists a closed form always?

*

*[I think no, but can't support my contradiction]


*Can taylor series be always used like this, atleast for approximation?

*

*[I think it always can be]



and similar ones, can somebody help?
 A: *

*Does integral always exist?


No. There are non-integrable functions. Integrability was a major reason to switch from Riemann integral to Lebesgue. However, continuous functions always have a primitive, so it's rather a theoretical concern, though a very important one.


*

*Does there always exist a closed form?


No. And you just pointed an example. Some such "bad" integrals get a name, when they are generally useful, and this give rise to a whole zoo of special functions. Yours above is called the sine integral.


*

*Can Taylor series always be used?


Only if the integrand can be expressed (easily) as a Taylor series. Often the series has no simple form, so it's just replacing a difficult problem with another as difficult. However, special function may often be defined as series expansion. And some functions are not analytic at all ($\exp (-1/x)$ is a classical example, around 0, and with Borel's lemma it's easy to get other ones, but see also here).
A: *

*The Lebesgue Integrability Condition says that any bounded, almost-everywhere continuous function on a compact interval is Riemann integrable.  In particular, this means that, say, $$\int_1^x \frac{\sin t}{t} \, dt$$
is a well-defined function.  In fact, in this particular case the integral from zero exists also, because the function
$$f(t) := \begin{cases}
\displaystyle\frac{sin(t)}{t} & \text{ if } t \neq 0 \\
1 & \text{otherwise}
\end{cases}$$
is continuous everywhere.  This function is sometimes called $Si(t)$ (for Sine integral), and you can see a graph here:  http://www.wolframalpha.com/input/?i=plot+integral+sin%28t%29%2Ft+dt

*There is no closed form for this integral in terms of elementary functions.  In fact that's the case for most integrals.

*Yes, you can commute integrals and sums, but there are certain convergence issues to worry about.
A: I don't know who told you $\int \dfrac{\sin x}{x}\; dx$ doesn't exist.  It does, and it even has a name: ${\rm Si}(x)$.  But it is not an elementary function.


*

*Every continuous function has an antiderivative.  

*Not every elementary function has an elementary antiderivative.  There is a well-developed theory behind this: see Risch algorithm.  "Closed form", on the other hand, is a nebulous concept: it may depend on whether somebody cared enough about this integral to give it a name.

*The Taylor series of any analytic function can be integrated term-by-term to give the Taylor series for an antiderivative.

