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Find the Maclaurin series of xcos(x^2).

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well, you know that

cos(x) = 1 - x^2/2! + x^4/4! - ...

so,

cos(x^2) = 1 - x^4/2! + x^8/4! - ...

Now just multiply that by x.

See

http://www.wolframalpha.com/input/?i=x+cos(x%5E2)

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