Help with an integral, yet again.

[DrKareem]

Hi, the integral I'm trying to solve is:

$$\int \frac{1}{1+\sin^2{\theta}}d\theta$$

From 0 to pi, which is the same as half the same integral over the unit circle.

I changed the sine squared into 1-cos[2x] and then expressed cos[2x] as half z^2 +(z-1)^2 and i finally got it in this form:

$$\frac{1}{i} \int \frac{z}{-z^4+z^2-1} dz$$

I found out the poles to be -1/2 +- i sqrt(3)/2 which happen to be on the countour path, and in that case i don't know how to apply the residue theorem, if it could be applied that is.

Edit: The answer is given in the book to be pi/sqrt(2)

 

[Gib Z]

Let sin(x)=u and express the entire integral in terms of u. the denominator is just 1+u^2, and solve and sub to dtheta to du, we get

$$\int \frac{\sqrt{1-u^2}}{1+u^2} du$$. Trig sub.

 

[Gib Z]

Ahh nvm my last post..I must have done something wrong because when you check it on www.calc101.com it dusnt gimme the same thing...But I don't undersatnd how you textbook gives you a constant for an indefinite integral..

BTW: The answer isn't nice..
Your textbook is correct if this is pi:

 

[DrKareem]

Okay I see where you're going here, it should be solved using that method. However, i took the integral from a chapter on the residue method in integration. So I'm trying to solve it using that method, and explained how i got stuck.

 

[Gib Z]

I have no idea what the theorem is :D

Edit: Method, not theorem

 

[joob]

your sub to cos(2x) is fine, and then exp's. what you then want to do is substitute $$z=e^{2 i \theta} thus dz=2i z d\theta$$. Your substitution was a little funky. thus $$cos(2 \theta) = \frac{z+ (1/z)}{2}$$.

 

[joob]

youll get a slightly easier integral which you can just take the residue of for the answer

 

[DrKareem]

Oh my God that was so trivial. Thanks for the help guys.

Edit: Actually it IS the residue THEOREM not method :P

 

[thebuttonfreak]

cool problem, shows the awesome power of the residue theorem!

 

[DrKareem]

Yes awesome indeed. I just started studying this and it looks really nice, although some aspects of it are hard to grasp. I guess i should be practicing even more...

 

[complexPHILOSOPHY]

Practice makes permanant!