Integral of a Trig FunctionDate: 12/20/95 at 20:2:34 From: Anonymous Subject: Calculus F(x)=Integral from -1 to 1 of 1/sin(x) Date: 6/20/96 at 9:49:19 From: Doctor Jerry Subject: Re: Calculus Since 1/sin(x) = csc(x) has a known antiderivative [ln(tan(x/2))], I'm assuming your question concerns this improper integral. Moreover, I'm assuming you don't want the Cauchy principle value of this integral, which, because of symmetry, would be zero. With these assumptions, the above integral converges if both the integrals Integral from -1 to 0 of 1/sin(x) and Integral from 0 to 1 of 1/sin(x) converge. Let's look at the second. Since for small positive x, sin(x) is very much x and int_0^1 (1/x) dx diverges, the given integal also diverges. Hence, the orginial integral diverges. More directly, int_a^1 csc(x) dx = ln(tan(1/2))-ln(tan(a/2)) As a -> 0, the right side becomes infinite. -Doctor Jerry, The Math Forum |
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