Tips for Integrating FunctionsDate: 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/ |
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