Differentiable functions without an antiderivative

Question

Specifically, why is there no antiderivative, or any possible method of integrating (except numerically) say $\;e^{\csc(x)}$?

(I don't have my computer handy right now so I cant format the formula, sorry about that!)

Answer 1

It's kind of similar to as saying: Why can we foil two linear terms to become a quadratic, but not any given quadratic can be factored into (real) linear terms (anti Foil, if you wish). That's just how it is. The far majority of continuous functions do not have an anti derivative in terms of elementary functions (that's I think what you mean here), and thus we need to resort to numerical methods to find area under the curve, or whatever the integral stands for.

Answer 2

I believe a Real valued function of a Real variable satisfies $\int f'=f$ iff f is absolutely continuous. The Cantor function is a standard example of a function that is not absolutely continuous. Since it is a.e. 0, it integrates to 0, so it doesn't have an antiderivative.

Edit: If f is Riemann-integrable, i.e., bounded and with a set of discontinuities of finite measure, then it has an antiderivative F. Specifically, $$ F= \int_0^x f(t) dt $$.

By the FTC: $F'(x)=f(x)$.