1. Hard Integration

Hello all
Could any one tell me which technique is suitable to solve this integration ? i will be thankful .
Best regards

2. Re: Hard Integration

Multiply both numerator and denominator by $\displaystyle \sqrt{tan(x)}- \sqrt{cot(x)}$.

3. Re: Hard Integration

I did this but no reduction , if you have a final result with steps i will thankful if you can give it to me :-)
Best regards

4. Re: Hard Integration

Are you looking for the integral defined over real numbers or over complex numbers? Because over real numbers, the function is not defined over large chunks of the reals. Over complex numbers, the problem becomes a little different.

Over the reals, you can do a small simplification by multiplying top and bottom by $\sqrt{\sin x \cos x}$. This will give you:

$\displaystyle \int \dfrac{\sqrt{\sin x} + \sqrt{\cos x}}{\sin x + \cos x}dx$

But, over the complex numbers, this simplification is not quite as straightforward. I am not sure I know a good technique for solving this problem.

5. Re: Hard Integration

Thanks Slip for replay
Also i did this modification , but still unsolvable :-(

Any idea ? in any domain , real , complex , etc
Best regards