-Substitution
Along with integration by parts, the -substitution is an integration technique that is frequently used for integrals that cannot be directly solved. The procedure is as follows:
(i) Find the term to be substituted for, and let that be
(ii) Find in terms of
(iii) Substitute and into the expression.
(iv) Integrate with respect to and then substitute back into the result.
For instance, say we have an integral of the form The process of solving this using -substitution is as follows:
(i) First we find the part to replace with In this case letting would be appropriate.
(ii) Then we find by differentiating both sides of
(iii) Now, replacing with and with gives
(iv) Finally, we integrate with respect to and substitute back into the result to obtain where is the constant of integration.
Observe that differentiating the result using differentiation of composite functions gives As you get used to -substitution, you will find out that integration via -substitution is the exact opposite of differentiation of composite functions, but let's just stick to the basics for now.
Contents
Integration -substitution - Given
For starters, it might not be so easy to determine for which part of the expression should be substituted. So, for our first step, we will start with problems where the substitution is given. As you walk through these examples, you will be gaining some sense in deciding which part of the integral to substitute for.
Using the substitution find
From we have Hence the given expression is equivalent to where is the constant of integration.
Using the substitution find
From we have Hence the given expression is equivalent to where is the constant of integration.
Using the substitution find
From we have Hence the given expression is equivalent to where is the constant of integration.
Using the substitution find
From we have Hence the given expression is equivalent to where is the constant of integration.
Integration -substitution - Definite Integrals
When it comes to integration using -substitution of definite integrals, we need to keep one more thing in mind: the integration interval must be changed to the interval of that corresponds to the given interval of Let's try the definite integrals of the functions we have integrated above.
Using the substitution find
From we have Hence the given expression is equivalent to
Using the substitution find
From we have Hence the given expression is equivalent to
Using the substitution find
From we have Hence the given expression is equivalent to
Using the substitution find
From we have Hence the given expression is equivalent to
Integration -substitution - Problem Solving - Basic
Integration -substitution - Problem Solving - Intermediate
-substitution is a great way to simplify integrals. It is a technique used in many other forms of integration such as integration by parts and the infamous trig sub.
-substitutions take two general forms, where or . Note that the chain rule for differentiation is basically equivalent to -sub for integrals.
Evaluate .
Let and . Then think of the as a "variable" and substitute it for Doing a -sub (the same thing as a -sub but with a different variable), we have Now, integrating by parts, Note that and don't mean differentiating with respect to and , but is the derivative of and is the derivative of . So, here we let and , then and both the integral and derivative of are
Hence, can be rewritten as For the last steps of our - and -substitutions, we must re-substitute for and and . Then we can rewrite as
Integration -substitution - Ln |f|
Now we will discuss a particular but frequently used form of -substitution, which is the form. Consider the integral of the form We substitute so that . Then is equivalent to where is the constant of integration. Hence we can easily compute the integrals of form , or we can manipulate expressions to turn them into that form.
Evaluate
Let Then we have and it follows that where is the constant of integration.
Evaluate
Let Then we have and it follows that where is the constant of integration.
Evaluate .
We have where is the constant of integration.
Evaluate .
We have where is the constant of integration.