# Tricky Integral

Hey

hope anybody can help me with a tricky integral ( i should check if it exists):

$$\int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{sin(x)}*cos^2(x)} dx$$

And i have really no idea where to start!

thanks

Dick
Homework Helper
You should definitely check if it exists before you work too hard on it. What's the behavior like at x=0 and x=pi/2?

LCKurtz
Homework Helper
Gold Member
You aren't going to be able to find a "simple" antiderivative. Maple gives a complicated answer using Elliptic functions.

So you mean, because it does not exist for Pi/2 it does not exist at all? so simple?

And yeah, i know i did check it with maple too!

vela
Staff Emeritus
Homework Helper
No, it's not that simple. You have to look at the behavior of the function around the singularities at x=0 and x=π/2. The integral can still converge is the function doesn't blow up too quickly around those two points.

ideasrule
Homework Helper
So you mean, because it does not exist for Pi/2 it does not exist at all? so simple?

That's not necessarily true. Consider the integral of 1/sqrt(x) from x=0 to 1. The integral exists, even though 1/sqrt(x) itself does not exist at x=0.

But if I try the limit x->Pi/2 i still get inf?

Dick
Homework Helper
But if I try the limit x->Pi/2 i still get inf?

The easiest way to see the problem at pi/2 is to expand your integrand in a Taylor series around x=pi/2 and realize the integrand looks like 1/y^2 where y=pi/2-x.

so you mean because the taylor series at x=pi/2 --> 1/(pi/2-x)^2 which is 1/0 at pi/2 and therefore not defined the integral does not exist? if yes, can you please tell me why? thx

Dick