Integration by Parts

First let’s take a look at the following.

So, that was simple enough.  Now, let’s take a look at,

To do this integral we’ll use the following substitution.

Again, simple enough to do provided you remember how to do substitutions.  By the way make sure that you can do these kinds of substitutions quickly and easily.  From this point on we are going to be doing these kinds of substitutions in our head.  If you have to stop and write these out with every problem you will find that it will take you significantly longer to do these problems.

Now, let’s look at the integral that we really want to do.

If we just had an x by itself or  by itself we could do the integral easily enough.  But, we don’t have them by themselves, they are instead multiplied together. 

There is no substitution that we can use on this integral that will allow us to do the integral.  So, at this point we don’t have the knowledge to do this integral. 

To do this integral we will need to use integration by parts so let’s derive the integration by parts formula.  We’ll start with the product rule.

Now, integrate both sides of this.

The left side is easy enough to integrate and we’ll split up the right side of the integral.

Note that technically we should have had a constant of integration show up on the left side after doing the integration.  We can drop it at this point since other constants of integration will be showing up down the road and they would just end up absorbing this one.

Finally, rewrite the formula as follows and we arrive at the integration by parts formula.

This is not the easiest formula to use however.  So, let’s do a couple of substitutions.

Both of these are just the standard Calc I substitutions that hopefully you are used to by now.  Don’t get excited by the fact that we are using two substitutions here.  They will work the same way. 

Using these substitutions gives us the formula that most people think of as the integration by parts formula.

To use this formula we will need to identify u and dv, compute du and v and then use the formula.  Note as well that computing v is very easy.  All we need to do is integrate dv.

So, let’s take a look at the integral above that we mentioned we wanted to do.