Question

What is the integral of `f(x)g(x)`?

Answer 1

There is no simple product rule for integration...

Explanation:

This appropriate and understandable question is almost certainly inspired by the product rule for differentiation, which tells us:

`(f(x) * g(x))' = f'(x) g(x) + f(x) g'(x)`

Unfortunately there is no such simple rule for integration.

For example, if `f(x) = 1/x` and `g(x) = e^x` then we have:

`int \ f(x) \ dx = ln x + C`

`int \ g(x) \ dx = e^x + C`

but

`int \ f(x) g(x) \ dx = Ei(x) + C`

where `Ei(x)` (the exponential integral) is not even an elementary function.