How Do I Integrate a WKBJ Semi-Classic Integral with a Square Root?

[karkas]

Homework Statement

I am having a problem integrating in a WKBJ semi-classic integral. Well it's this : I have to integrate

$\int_{0}^{\sqrt{m}E}\sqrt{E-\frac{x}{\sqrt{m}}}dx$

Homework Equations

Actually I don't have that much experience at integrating, so could you somehow show me how to integrate when you have a square root? Step by step this particular one, for example.

The Attempt at a Solution

I have tried setting the square root equal to a variable, t, and saying that the integral goes like
$\int_{0}^{\sqrt{m}E}t^2dt$ but it didn't seem to work out later on, plus I am almost sure this isn't correct.

 

[turin]

... could you somehow show me how to integrate when you have a square root?

I would just look it up in a table, as I usually do. I'm sure there's some trick that I was taught in calc 2, but you know, in my experience, most of those tricks are almost never useful anywhere besides a calc 2 test. And for such a simple integral, you can definitely find it in a table. Any integral of a squareroot of a 2nd order polynomial will be in even a modest table of integrals.

 

[turin]

I have tried setting the square root equal to a variable, t, and saying that the integral goes like
$\int_{0}^{\sqrt{m}E}t^2dt$ but it didn't seem to work out later on, plus I am almost sure this isn't correct.

OK, now I feel dumb. Yes, that is such an easy substitution. You just screwed up your limits. I'm guessing that you defined t as the squareroot. So, what is t when x=0 and what is t when x=\sqrt{m}E? Also, I think you get some additional factor.

 

[Brian_C]

That integral can be solved with a simple substitution. Hint: look at the quantity under the radical sign.

 

[karkas]

Is this what I should be getting from a table?

When i need to integrate $\int (ax+b) dx$ I set the square root equal to S and proceed to $\int_{0}^{\sqrt{m}E}S dx=\frac{2S^3}{3a}$ if $S=\sqrt{ax + b}$?

 

[Redbelly98]

Yes, though you probably meant to say

$\int \sqrt{ax+b} \ dx$

for the integral.

 

[karkas]

Yes indeed, my mistake! Well thanks for the help, I will work on it now :)