This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get something like:
$$ f(x) = h_1(t) |_{a}^{x} - \int_a^x g_1(x)$$
by repeating this process, you could end up with:
$$f(x) = \left( \sum_{k=0}^n h_k(t)|_a^x \right) + \int_a^x g_n(x) $$
and if you can bound $\int_a^x g_n(x)$ by something small then you have an approximation for the integral, $f$, in terms of elementary functions. My questions are:
- Are there any instances in which something like this turns out to be a successful strategy?
- Could someone direct me toward a book or resource about symbolically approximating integrals? By symbolic approximation I mean finding a good approximation to an integral $f(x) = \int_a^x g(t) dt $ of an elementary function in terms of elementary functions. When I google this, I only get results about numerically approximating integrals.
