Real Integral Solutions: Exact vs Approximations

[darkside00]

Is is possible to solve every real integral and come up with an actual solution? perhaps we may have not just found the methods of doing so. Or is it a must to use approximations(series/sequences) to do so?
Or is there a way to reverse numerical numbers to come up with a function?

 

[Hurkyl]

It really depends on what you mean by "actual solution".

Taylor series aren't approximations, by the way. You can truncate the series after a finite number of terms to produce an approximation, but the Taylor series itself is an exact representation of an analytic function -- at least within its radius of convergence.

 

[darkside00]

right. I guess all series can be expressed as functions with n to infinite within its convergence.

So, perhaps the real question is could we establish functions as a non series, or do we need them?

 

[HallsofIvy]

I think you need to formulate a precise question. What do you mean by "establish functions as a non series"?

 

[FallenRGH]

I think what he means is to define a function analytically without using a series expansion, and I believe the answer you are looking for is that, short of defining a function as the integral of another function(for example, the logarithmic integral function), we currently cannot define some functions short of a series expansion.