Integration by Parts
Example 1 Evaluate the following integral.
,
Now that we’ve chosen u we know that dv will be everything else that remains. So, here are the choices for u and dv as well as du and v.
The integral is then,
Once we have done the last integral in the problem we will add in the constant of integration to get our final answer.
Example 2 Evaluate the following integral.
Solution
This is the same integral that we looked at in the first example so we’ll use the same u and dv to get,
Example 3 Evaluate the following integral.
Solution
For this example we’ll use the following choices for u and dv.
The integral is then,
In this example, unlike the previous examples, the new integral will also require integration by parts. For this second integral we will use the following choices.
So, the integral becomes,
Example 4 Evaluate the following integral.
Solution
So, unlike any of the other integral we’ve done to this point there is only a single function in the integral and no polynomial sitting in front of the logarithm.
The first choice of many people here is to try and fit this into the pattern from above and make the following choices for u and dv.
This leads to a real problem however since that means v must be,
In other words, we would need to know the answer ahead of time in order to actually do the problem. So, this choice simply won’t work. Also notice that with this choice we’d get that
Therefore, if the logarithm doesn’t belong in the dv it must belong instead in the u. So, let’s use the following choices instead
The integral is then,
