Calculus I using LiveMath

Chapter 5

Integration Techniques - Substitution (the Chain rule in Reverse)

You know now that if you differentiate f(x) the resulting function, f'(x) can be integrated to give back the original function (or at least one of the family of the original function). Since you know how to differentiate a large number of functions, you could obviously make a very large table of integration formulas to help you find anti-derivatives.

Before the advent of the computer one would purchase large books of Integral Tables and most calculus books today have abbreviated lists. Many new math software packages eliminate the necessity of these tables because they have libraries attached to their programs with these formulas or complicated algohryms to solve most integration problems. All you have to do is enter an integral, and ask the computer to give you the answer.

The problem, as noted earlier in the book, is the fact that these libraries take up large chunks of memory and disk space. The people who developed LiveMath made their program flexible in that the default notebook only has the basic necessities. If needed however, you can build Notebooks with large libraries. There are several of these notebooks that came with your LiveMath program. If you have a very fast and large computer, you may want to create large notebooks with all the Transformation rules and Integral tables included. For those who don't though, and for those who want to learn the techniques for deriving the anti-derivatives of many functions, the following sections will try to accommodate.

In most of the integration problems you will be looking at, the Auto Casing menu item has been turned off, so the arbitrary constant will not be generated. This is only to keep the examples uncluttered. Keep in mind that this constant will be important in many applications you will work with.

Integration by Substitution

Eariler, the Chain Rule for differentiating composite functions was introduced. The Substitution Rule for Integration is the inverse of this.

Recall, the Chain Rule is a method of differentiation that is used when there is a composite function. To solve functions in this form you take the derivative of the "inside" function which was labeled u and take it times the derivative of the "outside" function, f(u). When you integrate the resulting function, F(x), in other words, reverse the process, you look for a inside function which looks like it is the derivative of u. The trick in this method is recognizing u.

As you learn this method of integration, the Chain Rule will become more understandable. Many students find that they really come to understand the Chain Rule after they work through the Substitution Rule. The best way to introduce the rule is to look at an example.

#5.11 The Chain Rule in Reverse

To help with the understanding of this rule you will first take a function and find its derivative. Then you will work backwards by integrating this new function to get back to the original one.

Turn Auto Simplify on and Auto Casing off!

• Input the following function and the Derivative OP:
  
• Take the derivative:
  Result:
• After commuting the x the Prop looks like the second Prop below:
  

This is the function you want to integrate!

 

• After you input it as an Integral, try having LiveMath integrate it by selecting Simplify. LiveMath will not be able to do the integration directly with this command.
  

As you look at the integrand you will notice that it could be split into two functions:

You are looking for a function that is the derivative of another one. In this case it is pretty easy to see that the first one, 2x, is the derivative of the inner function on the right, x^2+2. This is the expression that you want to define as u!

Although this is an easy one to just look at and recognize, there are going to be functions where it will be difficult for you to determine the inside function. In these cases use the computer to quickly take the derivative of different inner functions and look to see if one is the derivative of another one (this will be shown in the second example).

If you set up your notebook in the following manner, you will find that it will provide a good work area to do these types of problems. You may want to create a Substitution Rule notebook for these problems.

The Name d will take "Total Derivatives" as opposed to "Partial Derivatives" like the Derivative Op that you have used up to this point. The difference will become clear as you work through these examples and study more advanced topics. Make sure you put a space between the d and the u
(or you can type d * u ).

Your goal is to put the integral in the form:

To do this you make the appropriate substitutions. Since you know what you want for as u, you need to determine du and also have it in the form of an equation that you can substitute into the integral.

• Copy x2+2 from the first Prop (to select, drag the mouse through it or double click on the plus sign), and paste it into the ? defining u. After this is done, find the total derivative of u by substituting u=x^2 +2 into the du Prop. Your notebook should look like the following:
  

It looks as though you should be able to put the integral into the form:

But there is a problem! When you look at the integral, you can see that it will be easy to do the u substitution, but the du substitution cannot be done. That is because 2x and dx are not next to each other. There are two ways of getting around this.

 

• 1) The first method is to change the original integral around so that 2x and dx are together. To do this, select the 2x and Commute it over with the Hand to the other side of the expression. If this doesn't work, re-input the integral.
  

There you have it, the integral is in a form just waiting for you to do the substitutions.

If you have trouble substituting for 2x dx after doing the Commute procedure, it may be because of how you input your integrand and the differential in the first place. If you pasted the integrand into the Op, it will look like the following after you do a Commute.

LiveMath puts parenthesis around the whole integrand, and you will not be able to select dx with any part of the integrand inside the parenthesis. If you try to substitute the du equation into the integral, it will not work. Try it. The second method below will show you how to get around this.

• Once you have the integral with 2x on the right side next to the differential, substitute du into the integral.
  Result:
• Now substitute u into the new Integral Prop by selecting it and moving it to x^2+2 on the right side of the equation (not to the Prop icon, otherwise it will be substituted into the expression on the left too, and you do not want this).
  Result:
• Now all you have to do is substitute u in again for the final answer!!
  Result:

Remember, you do not have Auto Casing on at this point, so a Constant of Integration is not part of this answer.

2) The second method will give you the same answer and will be the method that will be the easiest to use on most integrals, mainly because you do not have to manipulate the integrand before the substitution.

 

• INPUT: The integral any way you want, as long as it is separated from the differential.
  
• After choosing the same u and finding du, as you did before (you do not need to commute 2x this time), the next thing you want to do is isolate dx by itself so that it can be substituted into the Integral. After doing both of these operations your notebook should look like the following:
  
• Next substitute u into your new Integral by selecting it and moving it to x^2+2 on the right side of the equation (not to the Prop icon).
  Result:
• To achieve the final answer, substitute u again into this newest equation:
  Result:

Which is the same answer as in Part 1.

• Normally you would differentiate after you got the answer to verify it was correct.
  Result:

Which is the original function, somewhat different only in that LiveMath has changed its form according to its conventions.

5.12 Complicate the Solution (just to learn)

This example will be solved using two different methods. The goal here is to show how, by searching for the best u in the beginning, you can save yourself time in the long run. Both methods do the job and the first, although the longer one, will demonstrate a new way to do substitutions and for some problems will be the only way it can be done.

• Input the following function:
  

Since it is not evident which expression to make equal to u, try differentiating a couple of them.

Two that are evident are:
1)

2)

Differentiate both.

• 1) Result:
  2) Result:

They both look like good candidates.

Part A

Using:

You are now left with another integral that is somewhat better, but one that still cannot be simplified for the answer. From here, you would continue by making a second substitution. This time, it is a little easier to see that if you use u^2+1 as the inner function, the numerator would be just a constant away from being its derivative.

This time use w as your substitution variable:

Part B

Using:

This example demonstrates that picking the correct substitution from the start, not only can save you time, but can eliminate the necessity of using a second substitution. As you try this LiveMath skill out on other problems, realize there is not a right way to integrate. The more problems you work, the better you will get at choosing the best method, or methods, to use.