Homework Help: Integral of squareroot and exponetial

1. Sep 15, 2010

Mechdude

1. The problem statement, all variables and given/known data
is there a general method of integrating this type of integral:
$$\int \sqrt{x -k} e^{-bx}$$

2. Relevant equations

3. The attempt at a solution

$$x-k={u}^{2}$$
$$dx=2udu$$
$$\int u^2 \,{e}^{{-b\left( {u}^{2}-k\right) }}du$$
$$\int {u}^{2}\,{e}^{{-b\,{u}^{2}}+{b\,k}}du$$

and it seems worse than its starting point

Last edited: Sep 15, 2010
2. Sep 15, 2010

csopi

Actually it's not worse... You can finish the problem by integrating by parts:

$$\int u^2\,{e}^{-bu^2}=\int u\cdot u{e}^{-bu^2}$$

I dont know what happened but as I type my message, the browser won't process more tex-code, so I put it in raw form, but I hope, you can understand from it what I'm saying. So:

=u\cdot \frac{{e}^{-bu^2}}{-2b}+\frac 1{2b}\,\int {e}^{-bu^2}

And it's done, because the last integral can be calculated explicitly, see the Gauss-distribution at wikipedia.

3. Sep 18, 2010

Mechdude

$$=u\cdot \frac{{e}^{-bu^2}}{-2b}+\frac 1{2b}\,\int {e}^{-bu^2}$$