Where Did I Go Wrong in Determining the Normalization Constant?

[atomicpedals]

Homework Statement

Determine the normalization constant c in the wave function given by
\psi(x) = c cos(kx) exp[(-1/2)(x/L)² ]

Homework Equations

1=\int |\psi(x)|² dx

limits of integration being -inf to inf.

The Attempt at a Solution

I'm very much sure that my math is wrong, I'm very rusty with improper integrals.

1= \int |c cos(kx) exp[(-1/2)(x/L)²|² dx

= \int |c² cos²(kx) exp[-(x²/L²)| dx

it's at this point I start getting into trouble

= c² \int |cos²(kx) exp[-(x²/L²)| dx

= c² \int cos²(kx)dx \int exp[-(x²/L²)dx

= c² (lim((2kx+sin(2kx))/4k)) (\pi)^1/2/(1/L²)^1/2

I think I'm pretty solidly wrong by this point... where did I go wrong?

 

[LaissezDairy]

You can't break up an integral like that. Think about it.

\int x^2 dx = \int x * x dx = \int x dx \int x dx = x^4/4 ??

Also this identity might make it less painful for you:

cos(kx) = \frac{e^{ikx} + e^{-ikx}}{2} (Euler's formula)

 

[atomicpedals]

Ah, right... well at least I made it three steps in before totally going off the deep end. Still working on it though.

 

[atomicpedals]

So my sticking point mathematically really seems to be the

e^(-1/2)(x/L)2

This almost certainly simplifies down to something reasonably basic after being squared and/or integrated shouldn't it?

 

[LaissezDairy]

do you know what a gaussian integral is?

 

[atomicpedals]

Yep, Arfken is a life-saver!