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\centerline{Mathematics 36, section 5 -- AP Calculus}
\centerline{Discussion 4 -- Tables of Integrals}
\centerline{October 26, 1999}
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\noindent
{\it Background}
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It is an unfortunate fact of life that ``most'' functions given by 
formulas {\it do not\/} have simple antiderivatives.  For instance,
integrals as simple as  
$$\int e^{-x^2}\ dx\quad {\rm and}\quad \int \sin\left({1\over t}\right)\ dt$$
cannot be expressed in terms of the usual ``library of functions'' that
we have been using (that is, the functions discussed in Chapter 1 of our
text book).  The forms that {\it can be integrated\/} have been extensively
studied and large {\it tables of integrals} have been compiled
by systematically applying various substitution 
techniques, integration by parts, etc.  The 
back of your textbook contains a very short extract from one of these
tables covering some of the most common and useful forms.  Here is an
example illustrating its use.  For instance, suppose we wanted to 
compute
$$\int \sin^4(5x)\ dx$$
Scanning the table, we see that the entry that comes closest to this 
is number 17, with $n = 4$.  Let's make the substitution $u = 5x$ 
first to ``clean things up.''  Then we have
$$\eqalign{ {1\over 5}\int \sin^4(u)\ du 
&= {1\over 5}\left( -{1\over 4}\sin^3(u) \cos(u) +{3\over 4}\int \sin^2(u)\ du\right)\cr
&= {-1\over 20} \sin^3(u) \cos(u) + {3\over 20} \int \sin^2(u)\ du\cr}$$
Using number 17 again on the remaining integral, we obtain:
$$\eqalign{
&= {-1\over 20} \sin^3(u) \cos(u) + {3\over 20}\left( -{1\over 2}\sin(u)\cos(u) 
+{1\over 2} \int \ du\right)\cr
&={-1\over 20} \sin^3(u) \cos(u) - {3\over 40}\sin(u)\cos(u) 
+{3\over 40} u + C\cr
&= {-1\over 20} \sin^3(5x) \cos(5x) - {3\over 40}\sin(5x)\cos(5x) 
+{3\over 8} x + C\cr}$$
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\noindent
As in this example, to use a table of integrals effectively, you 
must be able to:
\item{1)} recognize the underlying form of the function you want to integrate,
and decide which table entry applies,
\item{2)} make substitutions or perform other algebraic
``maneuvers'' as necessary to bring the integrand to 
the form given in the table,
\item{3)} apply the table entry,
\item{4)} convert back to the original variable if you made a substitution 
in 2.

\noindent
The following discussion questions give some practice at this:
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\noindent
{\it Discussion Questions}
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For each of the following integrals, say which entry or entries from the 
table you need to evaluate, and use them to express the integral
as function of $x$:
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\item{1)}
$$\int \cos^5(2x)\ dx$$
\item{2)} 
$$\int (x^2 - 2x)e^{4x}\ dx$$
\item{3)} 
$$\int x\cos(4x^2)\sin(12x^2)\ dx$$
\item{4)} 
$$\int {3x+1 \over x^2 - 3x + 2}\ dx$$
\item{5)} 
$$\int {1\over e^x - 1}\ dx$$
(Hint: multiply top and bottom by $e^x$, then try to find your
form in the table.)
\item{6)} $$\int {\tan(\ln(x^2)) \over x}\ dx$$
\item{7)} $$\int \cos^4(3x)\sin^2(3x)\ dx$$
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\noindent
{\it Assignment}
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Group writeups due Monday, November 1.
\bye

