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

May 19, 2018

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:

$\left(f \left(x\right) \cdot g \left(x\right)\right) ' = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$

Unfortunately there is no such simple rule for integration.

For example, if $f \left(x\right) = \frac{1}{x}$ and $g \left(x\right) = {e}^{x}$ then we have:

$\int \setminus f \left(x\right) \setminus \mathrm{dx} = \ln x + C$

$\int \setminus g \left(x\right) \setminus \mathrm{dx} = {e}^{x} + C$

but

$\int \setminus f \left(x\right) g \left(x\right) \setminus \mathrm{dx} = E i \left(x\right) + C$

where $E i \left(x\right)$ (the exponential integral) is not even an elementary function.