Expert: Paul Klarreich - 3/7/2006

Question
What is the Integration of

((csc x)/(2(sin x + cos x)))

Answer
Hi, Ibraheem,
Subject:  Calculas

>> That's Calculus.

Question:  What is the Integration of

((csc x)/(2(sin x + cos x)))
----------------------------
To integrate:
    csc x
----------------- dx
2(sin x + cos x)

you have to get rid of one of those terms on the bottom.  And having cot x appear in one place with  csc^2 x on top wouldn't be bad, either, since one is a derivative of the other.

Multiply top and bottom by csc x:

      csc^2 x
--------------------- dx =
2 csc x(sin x + cos x)

      csc^2 x
------------------------- dx  =
2 (1/sin x)(sin x + cos x)


      csc^2 x
--------------------- dx =
2(1 + cos x/sin x)

  csc^2 x
------------ dx =
2(1 + cot x)

Let  u = cot x
du = - csc^2 x dx

{    -1
| -------- du
{ 2(1 + u)


=  - (1/2) ln |1 + u|

etc.