Lesson 5.5 - Substitution

Many integrals can be tough to solve from the get go. Most of the time, the Calculus student will have to manipulate such integrands using some algebraic techniques, namely substitution. Substitution, also known as U-Substitution, is one of the most simple techniques to solve for an integral.

Initially, you start off with an integral such as . When you cannot solve it outright, you will need to make a substitution where .

You may use any variable here, but popular notation is to use the variable "u". Upon choosing your substitution, you must derive both sides of the equation with their respected variables like so:

The next step is to take these actions upon the original integral.

After solving for the new integral, the next step is to substitute back the original parameters and solve for the limits if is a definite integral. Substitution, like many topics in math, is best shown through a series of examples...

Example 5.5-A

What you need to recognize here are the powers. The derivative of is which is of the same form ( ) as the numerator of the integrand. We know what to do, so let's put our new found substitution technique into action!

We can now substitute these into the integral...

One last step: The original integral is in the form of x, so we need to back substitute our parameters...

Example 5.5-B

Once again, identity the powers. The derivative of is which is of the same form of .

Now let's try substitution that isn't involving powers upon x...

Example 5.5-C