1. ## "3 hour intergral"?

My prof was teaching a lecture then one thing lead to another and he talk about the fact that $\int \sqrt{sin\;x}\;dx$ DNE but however, $\int \sqrt{tan\;x}\;dx$ does! He followed by saying, "you can try at home to compute this 3 hour long integral" Joke about the time, but I wondering how hard this really is?

2. Look into Elliptic Integrals

3. Originally Posted by polymerase
$\int \sqrt{tan\;x}\;dx$ does! He followed by saying, "you can try at home to compute this 3 hour long integral" Joke about the time, but I wondering how hard this really is?
Well, it depends of your integration skills.

By settin' $u^2 = \tan x$ you'll get another nasty integral, but it isn't if you work with some of make-up

4. Originally Posted by Krizalid
but it isn't if you work with some of make-up
Wut exactly do u mean by make-up?

5. I mean you can manipulate the integrand as you want to get a nice result.

The straightforward-nasty way leads to apply partial fractions

6. How I would attempt it is perhaps via the beta function....assuming you had limits of integration.

For instance,

$\int_{0}^{\frac{\pi}{2}}\sqrt{sin(x)}dx$

You could use $2\int_{0}^{\frac{\pi}{2}}sin^{2p-1}(u)cos^{2q-1}(u)du$

With p=3/4 and q=1/2, we get:

$\frac{B(3/4,1/2)}{2}\approx{1.2}$

Just a little tidbit. Thought you may be interested.

7. Originally Posted by polymerase
[snip]
$\int \sqrt{tan\;x}\;dx$ [snip] "you can try at home to compute this 3 hour long integral" Joke about the time, but I wondering how hard this really is?
How hard ....? Only one way to find out, sport ..... try it!

You'll see it's not hard, just a bit long.

And after you've tried it, you can try $\int \frac{1}{\sqrt{tan\;x}} \;dx$.

8. How about, $\int \sqrt{\sin \frac{1}{x} }~dx$?
(Hint: This integral is non-sense, why?)