Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The question is $ \displaystyle \int{ \frac{1-r^{2}}{1-2r\cos(\theta)+r^{2}}} d\theta$.

I know it will be used weierstrass substitution to solve but i did not have any idea of it.

share|improve this question
    
Quotient rule? I'm familiar with a quotient rule for differentiation, but not with a quotient rule for integration. Can you tell us what you mean here by "quotient rule"? –  Gerry Myerson Oct 17 '11 at 21:40
    
@GerryMyerson: i am sorry, my mistake.. it is actually involving weierstrass substitution.. –  DRN Oct 18 '11 at 4:38
add comment

2 Answers

up vote 3 down vote accepted

Apply the substitution $$\tan \frac{\theta}{2}=t.$$ Then use $\cos\theta=\frac{1-t^2}{1+t^2}$.

share|improve this answer
    
how to apply $\tan{\frac{\theta}{2}}=t$ ? –  DRN Oct 17 '11 at 14:41
    
Then $sec^2(\theta/2).1/2 d\theta=dt\Rightarrow d\theta=\frac{2dt}{1+t^2}$. Now transform everything into $t$. –  Tapu Oct 17 '11 at 14:46
    
@Norlyda, I have seen a similar question here (in this site) just within one hour...but unable to find it. It would be helpful if someone post link here. –  Tapu Oct 17 '11 at 14:54
    
the problems occurs when I try to factorize the denominator in order to use integration by partial fraction. do you have any idea? –  DRN Oct 18 '11 at 7:12
    
@Norlyda, No partial fraction will be necessary. Your integration will boil into $2(1-r^2)\int\frac{dt}{(1+r)^2t^2+(1-r)^2}$. The denominator is of the form $ax^2+b=a(x^2+b/a)$. So the formula $\int\frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}(\frac{x}{a})$ will be used. –  Tapu Oct 18 '11 at 8:56
show 1 more comment

There's a Wikipedia article about this technique: Weierstrass substitution.

Notice that what you've got here is $\displaystyle\int\frac{d\theta}{a+b\cos\theta}$. The factor $1-r^2$ pulls out, and $a=1+r^2$ and $b=-2r$.

share|improve this answer
    
the problems occurs when I try to factorize the denominator in order to use integration by partial fraction. do you have any idea? –  DRN Oct 18 '11 at 7:12
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.