Simple Rules for Differentiation

Integration by Substitution

Objectives:

Students will be able to

· Calculate an indefinite integral requiring the method of substitution.

· Calculate a definite integral requiring the method of substitution.

We have already learned how to integrate functions that arise from differentiation power function, logarithmic functions, and exponential functions. Today we will look at how to integrate functions that arise from using the chain rule to differentiate. The method that we will use to do this is called the method of substitution.

Basically . This can be verified using the chain rule on the right side of the equation. The way that we will use substitution is to replace g(x) with u. This will mean that will be equal to du. Thus we only need to calculate . Once that integral is calculated, we can replace u in that integral with the original function g(x) to complete the process of finding .

Now we just need to figure out how to determine u. This is something that comes with practice. Just like in algebra class, we need to be able to look at a function that has come by composing two function and be able to decompose that function into its constituent parts.

Integration by Substitution

where

Please note that g is continuous and differentiable and f(u) is continuous at all points u in the range of g.

Example 1: