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Integral of a Trig Function

Date: 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

Associated Topics:
High School Calculus

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