This certainly isn't a complete list, but it is a few of the integrals you're likely to encounter.  For other integrals, you can either use a table or Wolfram's on-line Integrator.

 

Basic integrals:
     

Logarithmic & exponential integrals:
     

Trig integrals:
     

More complicated logarithmic integrals:
     
 

More exponential integrals: