Solving the Beta Function Problem: Integrating (2-x)/(2+x)^(1/2) from -2 to 2

[MisterMan]

Homework Statement

\int_{-2}^{2}\left(\frac{2-x}{2+x}\right)^{1/2}\hspace{1mm}dx

Homework Equations

B(p,q) = \int_0^{\infty}\frac{y^{p-1}}{(1+y)^{p+q}}\hspace{1mm}dy

The Attempt at a Solution

I am completely stuck on this one, just a total mental block. The answer in the book says:

"First put u = x + 2. Then put u = 4t. Answer = 2\pi"

I'll take you through what I done, but I got nowhere near the Beta function form I require:

u = x + 2\hspace{1mm}=>\hspace{1mm}4 - u = 2 - x

Also: du = dx. So:

\int_{0}^{4}\left(\frac{4-u}{u}\right)^{1/2}\hspace{1mm}du

So now I thought to do the second part ( set u = 4t ):

u = 4t => du = 4 dt

4\int_{0}^{1}\left(\frac{4-4t}{4t}\right)^{1/2}\hspace{1mm}dt

And I take out four as a common factor:

4\int_{0}^{1}\left(\frac{1-t}{t}\right)^{1/2}\hspace{1mm}dt

I don't have a clue on how to proceed, it is nowhere near the equation I need, I'm not sure how to get rid of the square root, or even "turn it upside down", but even then that doesn't work because I have 1 - t and I require something like 1 + t. I would really appreciate any help on this, thank you for your time.

 

[jackmell]

\int_0^1 \left(\frac{1-t}{t}\right)^{1/2}dt=\int_0^1 \left(\frac{1}{t}-1\right)^{1/2}dt

Now, you want the limits to go from zero to infinity. So looking at the integrand, what would I have to let say v=f(t) be so that when I plug in t=0 I get infinity but when I plug in t=1 I get zero?

Also, look at Wikipedia on the beta function to get familiar with it.

 

[jegues]

what would I have to let say v=f(t) be so that when I plug in t=0 I get infinity but when I plug in t=1 I get zero?

I think you mixed the two boundaries of the integral up.

When you plug in t = 0 into v you want 0, and similarly when you plug in t = 1 into v you want infinity.

Because the resulting boundaries you want are from 0 to infinity, right?

 

[jackmell]

Just let:

v=1/t-1 and work it through. Then switch the limits of integration.

 

[MisterMan]

Just let:

v=1/t-1 and work it through. Then switch the limits of integration.

Your solution doesn't make sense. It doesn't give me the limits I need and it isn't in the form I need it either.

 

[Raskolnikov]

4\int_{0}^{1}\left(\frac{4-4t}{4t}\right)^{1/2}\hspace{1mm}dt

And I take out four as a common factor:

4\int_{0}^{1}\left(\frac{1-t}{t}\right)^{1/2}\hspace{1mm}dt

[STRIKE]Wouldn't you be factoring out 4/4 = 1? So wouldn't the factor outside still be 1? Or am I just being silly here...Anyway, I hope this helps: [/STRIKE]
EDIT: yep...I was just being silly. I realized where your 4 came from finally haha.

4\int_{0}^{1}\left(\frac{1-t}{t}\right)^{1/2}\hspace{1mm}dt

= 4\int_{0}^{1}\left(t^{-1/2}(1-t)^{1/2}\right)} \ dt

= 4\int_{0}^{1}\left(t^{1/2 \ - \ 1}(1-t)^{3/2 \ - \ 1}\right)} \ dt

= 4B(1/2, \ 3/2)

= 8\int_{0}^{\pi / 2}\left(sin(\theta)^{2(1/2) \ - \ 1}cos(\theta)^{2(3/2) \ - \ 1}\right)} \ d\theta

= 8\int_{0}^{\pi / 2}\left(sin(\theta)^{0}cos(\theta)^{2}\right)} \ d\theta

= 8[.5(\theta + sin(\theta)cos(\theta))|_{0}^{\pi/2}

= 4(\pi /2)

= 2\pi.

 

[Raskolnikov]

By the way, read http://en.wikipedia.org/wiki/Beta_function for a list of the different forms of the beta function.

 

[MisterMan]

Thanks Raskolnikov, I've gotten so frustrated with that problem over the last day. I seemed to be too obsessed with getting the problem in the form:

B(p,q) = \int_0^{\infty}\frac{y^{p-1}}{(1+y)^{p+q}}\hspace{1mm}dy

That I took no notice of it being any in other form. ( Actually I'm a little confused why it is in a different form to the one I described initially, it was in a section dedicated to those types of Beta functions). Anyway, thanks everyone for your help :)