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Note : Of all the notes that I’ve written up for download, this is the one section that is tied to the book that we are currently using here at Lamar University.  In this section we discuss using tables of integrals to help us with some integrals.  However, I haven’t had the time to construct a table of my own and so I will be using the tables given in Stewart’s Calculus, Early Transcendentals (6th edition).  As soon as I get around to writing my own table I’ll post it online and make any appropriate changes to this section.

 

So, with that out of the way let’s get on with this section.

 

This section is entitled Using Integral Tables and we will be using integral tables.  However, at some level, this isn’t really the point of this section.  To a certain extent the real subject of this section is how to take advantage of known integrals to do integrals that may not look anything like the ones that we do know how to do or are given in a table of integrals.

 

For the most part we’ll be doing this by using substitution to put integrals into a form that we can deal with.  However, not all of the integrals will require a substitution.  For some integrals all that we need to do is a little rewriting of the integrand to get into a form that we can deal with.

 

We’ve already related a new integral to one we could deal with at least once.  In the last example in the Trig Substitution section we looked at the following integral.

 

 

 

 

At first glance this looks nothing like a trig substitution problem.  However, with the substitution  we could turn the integral into,

 

 

which definitely is a trig substitution problem (  ).  We actually did this process in a single step by using , but the point is that with a substitution we were able to convert an integral into a form that we could deal with.

 

So, let’s work a couple examples using substitutions and tables.

 

Example 1  Evaluate the following integral.

                                                              

Solution

So, the first thing we should do is go to the tables and see if there is anything in the tables that is close to this.  In the tables in Stewart we find the following integral,

                                 

 

This is nearly what we’ve got in our integral.  The only real difference is that we’ve got a coefficient in front of the x2 and the formula doesn’t.  This is easily enough dealt with.  All we need to do is the following manipulation on the integrand.

                    

 

So, we can now use the formula with .

                               

 

Example 2  Evaluate the following integral.

                                                        

Solution

Going through our tables we aren’t going to find anything that looks like this in them.  However, notice that with the substitution  we can rewrite the integral as,

                                           

and this is in the tables.

                               

 

Notice that this is a formula that will depend upon the value of a.  This will happen on occasion.  In our case we have  and  so we’ll use the second formula.

 

                               

 

This final example uses a type of formula known as a reduction formula.

 

Example 3  Evaluate the following integral.

                                                              

Solution

We’ll first need to use the substitution  since none of the formulas in our tables have that in them.  Doing this gives,

                                                   

 

To help us with this integral we’ll use the following formula.

                                          

 

Formulas like this are called reduction formulas.  Reduction formulas generally don’t explicitly give the integral.  Instead they reduce the integral to an easier one.  In fact they often reduce the integral to a different version of itself!

 

For our integral we’ll use .

                                        

 

At this stage we can either reuse the reduction formula with  or use the formula

                                                    

 

We’ll reuse the reduction formula with  so we can address something that happens on occasion.

                 

 

Don’t forget that .  Often people forget that and then get stuck on the final integral!

 

There really wasn’t a lot to this section.  Just don’t forget that sometimes a simple substitution or rewrite of an integral can take it from undoable to doable.


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