Calculus Problem
[Prev][Next][Index][Thread]
Calculus Problem
As some of you have already stated, sin(3x^2) doesn't have an explicit antiderivative.
Obviously you can't do it by u substitution, because there isn't a derivative for
u (if u = 3x^2, which it has to in this case, then there must be an x outside of
the sine function to act as its derivative - then the antiderivative is simple).
You can't do it by parts: the function is a composition of two functions - it isn't
a product of two functions. I'd say the best way to do it is to find the Taylor
polynomial for sin(3x^2) and integrate that. With a sufficient number of terms you
can get a very good approximation (sufficient being a very high order - that'll take
a while by hand or using the TI-92, but Mathcad should do it). Of course, you shouldn't
always use a Taylor series centered around x = 0 (McLaurin series); center it between
your limits of integration. Also, you could come up with a general summation expression
from the series that would approximate the integral. Integrating only a 10th order
McLaurin series I came within .03 of the definite integral of sin(3x^2) from 0 to
1 (the integral that approximates the integral of sin(3x^2) being 81x^11/440 - 9x^7/14
+ x^3). This integral is really only good from about -1 to 1, because it's centered
at x = 0. However, you could find different Taylor polynomials at various centers
between your limits of integration and then add their respective sums to find a very
good approximation (or just leave it in polynomial form for a general function).
Free web-based email, Forever, From anywhere!
http://www.mailexcite.com