# Hint needed in integral

core1985
Member warned about posting without the homework template
Hello I have tried gaussian integrals does gaussian integrals have this general form formula? if not then weather i do integration by parts or what just needed a hint to solve it correctly

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## Answers and Replies

Homework Helper
Hello Core,

The template is there for a reason, don't erase it but use it; it will be to your benefit.

What is the question ? and what is the relationship between your first line and the second ?

core1985
I want to say in the pic I have tried many things here should I show the steps I tried?? just want a hint that how to correctly start this nothing more weather I substitute or use gaussian integral formula for expomentional that is sqrt(pie/a) then do integration by parts??

Homework Helper
My point is it seems you are trying to normalize the wave function $$\Psi(x,t) = A\, e^{-x^2/a^2} e^{-i\omega t} \sin kx$$ on the first line.
But the second line does not reflect that ( it says ##\ \sin (2kx) \ ## instead of ##\ \sin^2 (kx) \ ## ).

So :
What is the question ? and what is the relationship between your first line and the second ?

core1985
yes yes it is sin^2(kx) we can use 1-cos2(x)/2 formula here

core1985
but that nasty exponential how to handle that

core1985
it is liboff problem 3.15 I have found <p> that is zero but I stuck at A?

core1985
these steps I have tried now where is mistake??

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core1985
Now see here in this formula list there no formula for x^2 that's why I am stuck at this step needed a hint

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core1985
how to integrate E^x^2/a^2 sinkx

Mentor
yes yes it is sin^2(kx) we can use 1-cos2(x)/2 formula here
For the latter expression, if you mean ##\frac{1 - \cos^2(x)}2##, use parentheses around the terms in the numerator. What you wrote means ##1 - \frac{\cos(2x)}2##. In any case, ##\sin^2(kx) \ne \frac{1 - \cos(2x)}{2}##. You have to consider that k mulitplier.

core1985
thanks I am new to this website

Homework Helper
Maybe you want to check out number 6 here ?

Otherwise there is CRC handbook of chemistry and physics, or Abramowitz (7.4.6)

core1985
so if I use number 6 then can I change limits to 0 to infinity multiplied by 2 then It can be applied ?

core1985
one thing more can I change sin(kx) into exponentionals and then try to solve will it work or not??

Homework Helper
so if I use number 6 then can I change limits to 0 to infinity multiplied by 2 then It can be applied ?
IF the function is even (##\ f(x) = f(-x)\ ##) then yes.
one thing more can I change sin(kx) into exponentionals and then try to solve will it work or not??
You can give it a try...

core1985
what do you suggest now changing sin to exponential using euler formula or use this

core1985
but cos(kx) is even ?? so I can use this to solve this nasty integral

core1985
ok I am solving it by both methods and will tell you what I got

Homework Helper
but cos(kx) is even ?? so I can use this to solve this nasty integral
Yes you can
what do you suggest now changing sin to exponential using euler formula or use this
That would be the idea. But it doesn't look clean and quick to me, such a complex exponential...

After all, integrating ##\ e^{-x^2}\ ## alone already requires ingenious mathematical manipulating...

core1985
ok then I use cos formula but can normalization have ? e term? according to the formula number 6 means I can write exponential in normalization