Integration by Parts
Integration by Parts is a method of integration that transforms products of functions in the integrand into other easily evaluated integrals. The rule is derivated from the product rule method of differentiation. Recalling the product rule, we start with
We then integrate both sides
We then solve for the integral of f(x)g'(x)
Integration by Parts
This is the formula for integration by parts. It allows us to compute difficult integrals by computing a less complex integral. Usually, to make notation easier, the following subsitutions will be made.
Let
Then
Making our substitutions, we obtain the formula
The trick to integrating by parts is strategically picking what function is u and dv:
1. The function for u should be easy to differentiate
2. The function for dv should be easy to integrate.
3. Finally, the integral of vdu needs to be easier to compute than the integral of udv.
Keep in mind that some integrals may require integration by parts more than once. Let's do a couple of examples
(1) Integrate
We can see that the integrand is a product of two functions, x and ex
Let
Then
Substituting into our formula, we would obtain the equation
Simplifying, we get
Integration by parts works with definite integration as well.
(2) Evaluate
Let
Then
Using the formula, we get
Then we solve for our bounds of integration : [0,3]
Let's do an example where we must integrate by parts more than once.
(3) Evaluate
Let
Then
Our formula would be
It looks like the integral on the right side isn't much of a help. Let's try integrating by parts and see if we can make it easier.
Let
Then
Our second formula would be
Substituting into our original formula, we would have
Notice that the integral on the left hand side of the equation appears on the right hand side as well, so we can solve for it.
Simplified, we get