In our previous lesson, Fundamental Theorem of Calculus, we explored the properties of Integration, how to evaluate a definite integral (FTC #1), and also how to take a derivative of an integral (FTC #2).
In this lesson, we will learn U-Substitution, also known as integration by substitution or simply u-sub for short.
Composite Function Notation
U-Substitution is a technique we use when the integrand is a composite function.
What’s a composite function again?
Well, the composition of functions is applying one function to the results of another.
Ok, but how does that help us with integrating?
Well, the first thing we will need to do is to recognize that we are being asked to integrate a product of a function and it’s derivative, and it takes the form of a composite function. This idea should now look familiar to you…
…Chain Rule!
Remember when we studied the Chain Rule while taking derivatives?
Well, U-Sub is nothing more than the reverse of the chain rule!
The Reverse Chain Rule
Great!
But how do we do this?
We will need to make a change of variables.
What?
According to Paul’s Online notes, the essence of the substitution rule is to take an integral in terms of X’s and transform or change it into terms of U’s.
How?
Identifying the Change of Variables for U-Substitution
Well, the key is to find the outside function and the inside function, where the outside function is the derivative of the inside function!
Then we will make a suitable substitution that will simplify our integrand so that we can integrate, as illustrated in three easy steps below:
The Steps for U Sub
Please note, that no matter what technique we use, the goal is always the same…
…we are finding the antiderivative of the integrand!
Together will see 11 examples of integration by substitution for both Indefinite and Definite Integrals.
U Substitution Video
U Substitution Examples
U Substitution Examples covering both Definite and Indefinite Integrals
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