10.2 Usubstitution Indefinite Integralsap Calculus



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.

U-Substitution is a technique we use when the integrand is a composite function.

Making a Fast Switch: Variable Substitution - Indefinite Integrals - Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. This book offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams. It covers intermediate calculus topics in plain English, featuring in. Instructor What we're going to do in this video is get some practice applying u-substitution to definite integrals. So let's say we have the integral, so we're gonna go from x equals one to x equals two, and the integral is two x times x squared plus one to the third power dx.

What’s a composite function again?

Well, the composition of functions is applying one function to the results of another.

10.2 U-substitution Indefinite Integralsap Calculus Calculator

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

10.2 u-substitution indefinite integralsap calculus calculatorIntegralsap

10.2 U-substitution Indefinite Integralsap Calculus Definition

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:

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

10.2 U-substitution Indefinite Integralsap Calculus Integrals

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