How do you integrate #f(x)=(x^2+x+1)(x-1)# using the product rule?
Jim H. is correct; there is no product rule for integration but there is an integration method called, Integration by Parts , that is so closely related to the product rule that the product rule can be used to derive the method. The reference contains the derivation and I will not repeat it here but I will use the method to perform the integration.
NOTE: Normally, I would not use integration by parts; I would, perform the multiplication, simplify, and then integrate each term but you requested the use of the product rule, therefore, I will use the closest possible relative to the product rule.
The integration by parts formula is:
To apply the formula to the given factors, we let:
Substituting these 4 equations into the formula:
The remaining integral is done "term-by-term", using the power rule,
Please compare the above answer with the answer that one receives, if one just performs the multiplication:
This method does not require the tedious simplification that the other method requires.