Expert: Paul Klarreich - 7/14/2007

Question
I tried for atleast an hour on this question   

Using U substitution Find the Integral of (cot sqaured x) divided by (csc sqaured x).  

Answer
Questioner:   Ashley
Category:  Advanced Math
Private:  No
 
Subject:  Calculus II u substitution
Question:  I tried for atleast an hour on this question   

Using U substitution Find the Integral of (cot sqaured x) divided by (csc sqaured x).
..............................................
Hi, Ashley,

WARNING: VIEW THIS IN A FIXED-SIZE FONT, SUCH AS COURIER.

{ cot^2(x)
| --------- dx
} csc^2(x)

A lot of the time, it makes things easier to reduce things to sines and cosines, as in:


{ cos^2(x)/sin^2(x)
| ----------------- dx
} 1/sin^2(x)

{ cos^2(x) sin^2(x)
| ----------------- dx
} sin^2(x)    1

{
| cos^2(x) dx
}

Now there is standard 'trick' for  cos^2(x), also for  sin^2(x) if that comes up.  It's called the 'half-angle' identity:
          1 + cos(2x)
cos^2(x) = -----------
              2

Your integral is:

{ 1 + cos(2x)
| ------------- dx
}     2

x + sin(2x)/2
--------------
    2

etc.