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