Tips for Integrating Functions

Date: 05/27/98 at 23:10:32
From: Brian C.
Subject: Pointers/Tutorial on Integration

Hi!

I have been looking all over the Net for the past few weeks for a good 
tutorial on finding anti-derivatives, and haven't had much luck. There 
is an abundance of excellent sources on Derivative Calculus but none 
on Integral, so I am turning to you for help.

Unfortunately it seems there is not an easy formula for finding 
an Anti-Derivative, as there is for finding a derivative 
(f'(x)=(f(x+h)-f(x))/h). So if a good explanation would be too 
lengthy, even a pointer to a tutorial would be great.

Thanks for maintaining such a wonderful site, as it has helped me with 
numerous issues, not to mention it's just cool going "Ahhh, so that's 
why." =)

Brian Cowan

Date: 05/28/98 at 13:40:13
From: Doctor Rob
Subject: Re: Pointers/Tutorial on Integration

It is an unfortunate fact that while every familiar function of 
calculus has a derivative, not all of them have antiderivatives 
expressible in terms of those familiar functions. A classic example is 
the so-called elliptic integrals, which first appear when trying to 
find the arc-length of an ellipse. The arc-length s from x = 0 to 
x = a * sin(T) is given by:

   s = a * Integral sqrt(1 - k^2 * sin^2[t]) dt  from t = 0 to T

where k is the eccentricity of the ellipse. For ellipses that are not
circles, k is not 1, and this integral is not possible in closed form.
Other examples are the integrals of sqrt(sin(x)), e^(-x^2), and so on.

You are correct that there is no standard method for doing 
integration. Some classes of functions have such methods, however.

For rational functions, the idea is to decompose the rational function
into a polynomial plus a rational function whose numerator has smaller
degree than the denominator. Then this new rational function can be
split into simpler ones by factoring the denominator, and splitting it
up via the method of partial fractions. These can be integrated
individually.

For rational functions of trigonometric functions, the method is to 
use multiple angle formulas to express the function in terms of sines 
and cosines of a single angle t. Then the substitution z = tan(t/2) 
will reduce the function to a rational function in z. This reduces it 
to the previous case.

For more elaborate functions, and those of other types, a high degree 
of pattern-recognition and experience in choosing substitutions that 
will simplify the problem at hand are involved. Often when there is a 
common subexpression among two or more parts of the integrand, it is 
useful to substitute some new variable for it. When an expression of 
the form sqrt(a^2-t^2) is encountered, the substitution t = a*sin(z) 
will often help, or t = q*coth(z). When it is sqrt(a^2+t^2), the 
substutution t = a*tan(z) will often help, or else t = a*cosh(z).

Another very useful technique is integration by parts. When to use it 
is not, however, very clear. When the integrand is a product, one of 
whose parts is integrable, and the other of which has a relatively 
simple derivative, then integration by parts may be beneficial.

In summary, there are just a few useful techniques that are useful in
integration. They have to be applied imaginatively. Even so, some
functions will turn out not to be integrable at all.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/