Solving Exponential Integral with Pathlengths for Solids

[k_c]

Homework Statement

I'm working out different pathlengths for different solids but I'm stuck on the following integral:

Homework Equations

\int x*exp(-C*sqrt[1-x²/A²]) dx

exp = exponential func
sqrt = square root
C and A are constants

The Attempt at a Solution

I tried to work it out with the substitution method, where u = -C*sqrt[1-x²/A²] and du = C/A²x(1-x²/A²)^-1/2.

But it seems to be getting really complex afterwards, so I was wondering if I'm overlooking the simple approach for this integral?

 

[Whitishcube]

try integration by parts.

 

[k_c]

I tried it this way now:

u^2 = (1-x^2/a^2)
so du = -2x/a^2 dx and therefore x dx = -a^2/2 du

So the integral becomes: \int -a^2/2 exp^(-cu) du
And then i find the solution: a^2/(2c) exp^(-cu)

Which is a^2/(2c) exp^[-c sqrt(1-x^2/a^2)]


I think this should be correct but the wolfram mathematica integrator gives me a different solution, so if somebody could confirm my method/solution, that would be great..

 

[ehild]

I tried it this way now:

u^2 = (1-x^2/a^2)
so du = -2x/a^2 dx and therefore x dx = -a^2/2 du

Check du.

ehild

 

[hunt_mat]

For the integral:
<br /> \int x\exp (-c\sqrt{1-x^{2}/a^{2}})dx<br />
I use the substitute:
<br /> u^{2}=1-\frac{x^{2}}{a^{2}}<br />
Then:
<br /> xdx=a^{2}udu<br />
This makes the integral become:
<br /> a^{2}\int ue^{-u}du<br />
This I think is easy to compute, so I will leave that part to you.