Integral[0, pi/2] sqrt[cot(x)]/{sqrt[cot(x)] + sqrt[tan(x)]} dx

[Steven G]

\(\displaystyle \displaystyle \mbox{Evaluate: }\, \int_{0}^{\frac{\pi}{2}}\, \dfrac{\sqrt{\strut \cot(x)\,}}{\sqrt{\strut \cot(x)\,}\, +\, \sqrt{\strut \tan(x)\,}}\, dx \)

Hint: Using symmetry and addition will yield an integral whose integrand is 1. This integral with integrand of 1 will help evaluate the initial integral.

 

[Steven G]

\(\displaystyle \displaystyle \mbox{Evaluate: }\, \int_{0}^{\frac{\pi}{2}}\, \dfrac{\sqrt{\strut \cot(x)\,}}{\sqrt{\strut \cot(x)\,}\, +\, \sqrt{\strut \tan(x)\,}}\, dx \)

Hint: Using symmetry and addition will yield an integral whose integrand is 1. This integral with integrand of 1 will help evaluate the initial integral.

Does anyone want to try this?

 

[Deleted member 4993]

Let us change the problem to:

\(\displaystyle \displaystyle \mbox{Evaluate: }\, \lim_{\epsilon \to 0} \left[\int_{\epsilon}^{\frac{\pi}{2}- \epsilon}\, \dfrac{\sqrt{\strut \cot(x)\,}}{\sqrt{\strut \cot(x)\,}\, +\, \sqrt{\strut \tan(x)\,}}\, dx \right]\)

=\(\displaystyle \displaystyle \mbox{Evaluate: }\, \lim_{\epsilon \to 0} \left[\int_{\epsilon}^{\frac{\pi}{2}- \epsilon}\, \dfrac{1}{1 +\, \tan(x)}\, dx \right]\)

Eikes.....

Need to try it later......

 

[Deleted member 4993]

Let us change the problem to:

\(\displaystyle \displaystyle \mbox{Evaluate: }\, \lim_{\epsilon \to 0} \left[\int_{\epsilon}^{\frac{\pi}{2}- \epsilon}\, \dfrac{\sqrt{\strut \cot(x)\,}}{\sqrt{\strut \cot(x)\,}\, +\, \sqrt{\strut \tan(x)\,}}\, dx \right]\)

=\(\displaystyle \displaystyle \mbox{Evaluate: }\, \lim_{\epsilon \to 0} \left[\int_{\epsilon}^{\frac{\pi}{2}- \epsilon}\, \dfrac{1}{1 +\, \tan(x)}\, dx \right]\)

Eikes.....

Need to try it later......

Wolfram says the answer is pi/4.

What did you get?

 

[Steven G]

Wolfram says the answer is pi/4.

What did you get?

I got pi/4. You should use symmetry via a u-substitution. It is really a nice problem once you get it.

 

[Steven G]

\(\displaystyle \displaystyle \mbox{Evaluate: }\, \int_{0}^{\frac{\pi}{2}}\, \dfrac{\sqrt{\strut \cot(x)\,}}{\sqrt{\strut \cot(x)\,}\, +\, \sqrt{\strut \tan(x)\,}}\, dx \)

Hint: Using symmetry and addition will yield an integral whose integrand is 1. This integral with integrand of 1 will help evaluate the initial integral.

Nobody was able to do this problem? Just let u = x-pi/2 and beautiful things will happen. Try it you'll like it! (Except Denis who hates everything). Anyone who does it (no wolfram alpha) will receive an extra 10% (including Denis) in their weekly paycheck. BTW, when do I get my first check?