Examples:

1. ex:=x^n
exdiff := diff(ex,x)
exdiffint := int(exdiff,x)
exint := int(ex,x)
exintdiff := diff(exint,x)
simplify(exintdiff)

   

   Comments: We had to simplify the result of the integration followed by differentiation to get back to the same function.

2. Integration by parts

   e2:= exp(-a*x)*cos(b*x)
int2:= int(e2, x)
simplify(int2)

   

   Comment: Note that Maple knows how to integrate by parts. You might try doing this integral yourself and see how long it takes you compared to letting Maple do it for you.

3. Integration with a singularity

   e4 := (1+x^4-2*x^5) / (1-x^8)
inte4 := int(e4, x)

   

   Comment: This integrand has a singularity, the denominator approaches zero as x approaches 1. To do the integral, you must express the integrand in partial fractions. Once again, you might try doing this integral yourself using partial fractions and compare this to the time required to use Maple. You should start becoming convinced that Maple can help you a lot in homework exercises where calculus is required.