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 e^x

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