Taylor Series: Integral of sin(x^2)

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Gchem45

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Dec 11, 2006
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The problem I need help with is:
-Use a Taylor series to find a formula to calculate the integral of sin(x^2).

I know this problem isnt very difficult it just take a good deal of work. However, I'm still a little rusty on how to apply the Taylor series to functions that are outside the basic format given in textbook examples. All I need is to be pointed in the right direction with a format or a formula and how it applies to this problem.
 
You probably know the expansion for sin(x). Sub in x^2 for x

You get:

\(\displaystyle x^{2}-\frac{x^{6}}{6}+\frac{x^{10}}{120}-\frac{x^{14}}{5040}+\frac{x^{18}}{362880}{-}.....=x^{2}-\frac{x^{6}}{3!}+\frac{x^{10}}{5!}-\frac{x^{14}}{7!}+\frac{x^{18}}{9!}{-}...........\)

\(\displaystyle \L\\\sum_{n=1}^{\infty}{(-1)^{n+1}\frac{x^{4n-2}}{(2n-1)!}\)
 
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