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

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Answer 1 · 2018-05-19T18:20:04

There is no simple product rule for integration...

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.