Integration Trick

Date: 8/13/96 at 16:59:44
From: Anonymous
Subject: Integral of Csc (x)

f(x)=1/sin(x). What is the indefinite integral of f(x)?

I got:

F(x)=ln(abs(sin(x))/abs(cos(x)+1))+C

But I'm not sure if this is correct.

Date: 8/30/96 at 17:30:37
From: Doctor Jerry
Subject: Re: Integral of Csc (x)

I differentiated and simplified your answer, obtaining 1/sin(x).  This 
shows it is correct.

In a table I own, the formula is given this way:

integral of csc(x) dx = ln(abs(tan(x/2)))+c

I remember this integral from 1950, when I first learned calculus.  
I liked the trick to do the integration. Just take csc x and multiply 
it by 1, where 

1 = (csc x + cot x)/(csc x + cot x)

Then, note that, except for a minus sign, the numerator is the
derivative of the denominator.  So, one gets -ln(csc x + cot x).
This can be written in several different ways.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/