Expert: Paul Klarreich - 4/7/2006

Question
My name is Megan and I can not figure out how to integrate cos 4x/(1+2 sin4x)^4 dx
I have tried using substitution letting u=1+2sin4x, however I end up with -1/24(1+2sin4x)^3 +c which is not the correct answer

Answer
Hi, Megan,
Subject:  integration by substitution
Question:  My name is Megan and I can not figure out how to integrate cos 4x/(1+2 sin4x)^4 dx
I have tried using substitution letting u=1+2sin4x, however I end up with -1/24(1+2sin4x)^3 +c which is not the correct answer
-------------------------------------
That seems to be a legitimate way to do it.  It goes likes this:
{     cos 4x dx
| -----------------
} (1 + 2 sin 4x)^4

Let u = 1 + 2 sin 4x
du = 8 cos 4x dx

{   du
| ------
}  8u^4

1  {
_  | u^(-4) du
8  }

1 u^(-3)
- ------
8  -3
  1        1
- -- ----------------
 24 (1 + 2 sin 4x)^3

Looks OK to me.  Of course, sometimes the 'answer in the back of the book' is a different form, especially involving trigonometric functions.