Integration by Substitution

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Integration by Substitution

Introduction to this topic[edit]

This page is dedicated to teaching techniques for integration by substitution. For other integration methods, see other sources.

The first section introduces the theory. Next comes a demonstration of the technique; this is followed by a section listing the steps used in that demonstration. The last section is a series of clarifying examples.

Assumed Knowledge[edit]

  • Basic differentiation
  • Basic integration methods

To understand the theory:

Theory of Integration by Substitution[edit]

This area is covered by the Wikipedia article Integration by substitution. On this page we deal with the practical aspects.

We begin with the following as is described by the Wikipedia article

This can be rewritten as

by setting

The principle applied here is function of a function (Function composition) and the reverse of the chain rule. This is the basis of integration by substitution.
The key skill now is to identify what value we use for u and following the process to solution.

Integration by substitution[edit]

The objective of Integration by substitution is to substitute the integrand from an expression with variable x to an expression with variable where

Theory

We want to transform the Integral from a function of x to a function of u



Starting with




Steps

       (1) ie   
     (2) ie   
     (3) ie   
     (4) ie   Now equate with
     (5) ie   
     (6) ie   
     (7) ie   We have achieved our desired result

Procedure

  • Calculate
  • Calculate which is and make sure you express the result in terms of the variable u
  • Calculate
  • Calculate

Technique[edit]

Example 1[edit]

Let us examine this integral

The inner function is

The outer function is

Recognising this relationship we then move onto the following set of steps to process the inner function
NOTE: that the differential of is .

Now we substitute u and du into the original integral.

Then apply standard integral technique

And finally we substitute the value of u back into the equation

Example 2[edit]

Let us examine this integral

We can first rearrange the fraction to make it more familiar.

The inner function is

The outer function is

Next we assign u and du

But we have a problem! doesnt equal ! So we need to rearrange our formula for .

Now we can substitute u and du into the original integral.

Study the above substitution carefully. We moved the fractional component of du to the front as it represents a constant.

Now apply standard integral technique

Cleaning up this expression we have

And finally we substitute the value of u back into the equation

The Definite Integral[edit]

Consider the definite integral

By using the substitution

Now because we have limits, we need to change them with respect to u. Note the value of the limits.

Now we have a new definite integral to solve

The Steps We Applied[edit]

Let's now review the steps for integration by substitution.

Indefinite Integral Definite Integral
1. First identify that you have a function of a function. This skill comes with practice to identify candidates. First identify that you have a function of a function. This skill comes with practice to identify candidates.
2. Identify u and then find du that is appropriate for the expression. Identify u and then find du that is appropriate for the expression.
3. Change limits for definite integrals.
4. Integrate using normal techniques. Integrate using normal techniques.
5. Substitute back the values for u for indefinite integrals.
6. Don't forget the constant of integration for indefinite integrals.

Finding u[edit]

Lets look at more examples at finding u.

Example 1[edit]