1) The document demonstrates integration by parts using two examples. 2) In the first example, letting u = x and dv = e^x dx, the integral is solved to be x*e^x - ∫e^x dx. 3) In the second example, recursion occurs due to a second integration by parts. A math trick is used to remove the recursion by subtracting integrals and factoring terms until the solution is isolated.

1. Recall integration by parts:
∫ ∫ ∫
∫ ∫
Or (subtract ∫ from both sides):
∫ ∫
So if we are given:
∫ ( )
Let:
( )
( )
Then:
∫ ( ) ( ) ∫ ( )
Boner city, we have to do another integration by parts run, this time let:
( )
( )
So:

2. ( ) ∫ ( ) ( ) ( ) ( )∫ ( )
Confusing right? We have a bit of recursion going on, we need to nip this in the bud, let’s do a little math
trick, but first let’s clean this up a bit:
( ) ( ) ( )∫ ( )
( ) ( ) ( )∫ ( )
Here’s the trick, do you see something similar to the answer we have and the problem we were tasked
with? They have the same integral on both sides of the equality:
∫ ( ) ( ) ( ) ( )∫ ( )
Let’s subtract ( ) ∫ ( ) from both sides and see what we’re left with:
∫ ( ) ( )∫ ( ) ( ) ( )
( )∫ ( ) ( ) ( )
Now let’s multiply the both sides by :
( )∫ ( ) ( ) ( )
Factor the right side by :
( )∫ ( ) ( ( ) ( ))
Ah ha! We are almost there! Now let’s divide both sides by ( ) and see what we’re left with:
( ) ( )
∫ ( ) ( )
Viola! Isn’t math beautiful?