Saturday, October 3, 2009

Tricky definite integral

Evaluate the integral from -1 to 1 of cos(x)/(e^x + 1) dx.
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1 comment:

  1. DavidOctober 4, 2009 at 4:07 PM

    The answer is sin(1).

    Let f(x)=cos(x)/(e^x+1). Now, every real function is uniquely the sum of an even function e(x)=(f(x)+f(-x))/2 and an odd function o(x)=(f(x)-f(-x))/2. Since the integral of an odd function over the interval [-1,1] is 0, it suffices to integrate the even part.

    Now, it is easy to check that e(x)=cos(x)/2, and the integral from -1 to 1 of cos(x)/2 dx is sin(1).

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