Integral approximation project, what integrals should I choose?

Question

For a highschool project I've been asked to make a scientific investigation of some sort, I chose numerical methods to calculate integrals. I want to use multiple methods and therefore wonder what integrals I should choose to show the advantages and disadvantages of the different methods. My current list of methods to try are: Riemann sums, Taylor series and interchanging order of integration and summation (i.e. for the gaussian integral). Thanks in advance for any help