Substitution Rule

Mostly integrals are fairly simple and most of the substitutions are quite simple. The problems arise in correctly getting the integral set up for the substitution(s) to be done.  Once you illustrate how these are done it's simple to see what you ought to do, however the first time through these can cause problems if you aren't on the lookout for potential problems.

Example Evaluate following integrals.

      ∫ e^2t  + sec ( 2t ) tan ( 2t ) dt

Solution

This integral contains two terms in it and both will need the similar substitution. This means that we ought not to do anything special to the integral. One of the more common "mistakes" here is to break the integral and carry out a separate substitution on each of the part. It isn't really mistake although will definitely enhance the amount of work we'll have to do.  Therefore, since both terms in the integral utilizes the similar substitution we'll just do everything like a single integral by using the following substitution.

                     u = 2t                          du = 2dt⇒           dt = 1/2 du

Then the integral is,

∫ e^2t  + sec ( 2t ) tan ( 2t) dt = 1/2 ∫ e^u  + sec (u ) tan (u ) du

= 1 /2(e^u  + sec (u ))+ c

= 1/2 (e^2t  + sec ( 2t )) + c

Frequently a substitution can be utilized multiple times in an integral thus don't get excited about that if it happens.  Also note as well that since there was a  ½ in front of the whole integral there have to be a  1 /2 also in front of the answer from the integral.