Expert: Paul Klarreich - 4/24/2007

Question
Hi there
I have a small problem
How do i integrate: ( cosx + sinx + 1 )/(1 + (sinx)^2)

??

Answer
Questioner:   iwinux
Category:  Calculus
 
Subject:  Integral
Question:  Hi there
I have a small problem
How do i integrate: ( cosx + sinx + 1 )/(1 + (sinx)^2)

************ USE COURIER FONT TO VIEW THIS.*********

There are three terms, and you will handle each one separately:

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

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

{      1
| -------------- dx
} 1 + (sin x)^2
. . . . . . . . . . . . . . . . . . .
The first one is a  u = sin x substitution.  du = cos x dx, and the integral becomes:

{    1
| -------- du
} 1 + u^2

That will be an arctan(u), etc, and you will back substitute.
. . . . . . . . . . . . . . . . .
The second can be written:

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

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

Now use  u = cos x.  This will also involve an arctan.  There will be a minus in there somewhere, and you will have to deal with a factor of  sqrt(2), but you can handle it.
. . . . . . . . . . . . . . . .
Third one:

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

This is not so easy.  Try this: Make the same transformation as in part 2:

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

Next multiply top and bottom by sec^2(x):

{    sec^2 x
| ------------- dx
} 2sec^2(x) - 1

Now, remembering that sec^2(x) is the derivative of tan x, AND that  sec^2(x) = 1 + tan^2(x):

{    sec^2 x
| ------------------- dx
} 2(1 + tan^2(x)) - 1

{    sec^2 x
| ------------------- dx
} 2 + 2 tan^2(x)) - 1

{    sec^2 x
| --------------- dx
} 1 + 2 tan^2(x))

Now let  u = tan x,  du = sec^2 x dx

{    1
| ---------- du
} 1 + 2 u^2

Now do this one like the others.  There will be a sqrt(2) in there, perhaps in a couple of places, but you can handle that.
. . . . . . . . . . . . . . .  
FULL DISCLOSURE STATEMENT.

I cheated.  There is a program on the web called 'The Integrator'.  (No, that is not the name of an Arnold Schwarzenegger movie.)  That program says the answer is:

arctan( sqrt(2) tan(x) )/sqrt(2)

Using this hint, I was able to figure a way.