Does anybody know the examples of indefinite or definite integral that can be done in the terms of elementary functions manually by a good first-year or second-year student, but which Wolfram Alpha (or Mathematica) evaluate not correctly? In other words, I want to find some tasks for mathematical test, where students cannot easily find the answer using wolfram and just rewrite it in their papers. Thanks in advance.
It is probably impossible. Set of functions known by 1-2 years students is constrained. Mathematica uses symbolic algebra system to transform integrals, and big repository with properties of functions.
For example for Hypergeometric Functions you have (218,254 formulas)!
Methods of calculations of integrals are explained on wolframalpha.com as step by step solutions for pro users. ($4.75/mo billed annually or $6 billed monthly)
Calculating integrals by computers is nowadays on level comparable to chess games. You have to talk with student individually.