# Hint needed in integral

1. Nov 18, 2016

### 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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2. Nov 18, 2016

### BvU

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 ?

3. Nov 18, 2016

### 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??

4. Nov 18, 2016

### BvU

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 :

5. Nov 18, 2016

### core1985

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

6. Nov 18, 2016

### core1985

but that nasty exponential how to handle that

7. Nov 18, 2016

### core1985

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

8. Nov 18, 2016

### core1985

these steps I have tried now where is mistake??

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9. Nov 18, 2016

### core1985

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

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10. Nov 18, 2016

### core1985

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

11. Nov 18, 2016

### Staff: Mentor

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.

12. Nov 18, 2016

### core1985

thanks I am new to this website

13. Nov 18, 2016

### BvU

Maybe you want to check out number 6 here ?

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

14. Nov 18, 2016

### core1985

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

15. Nov 18, 2016

### core1985

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

16. Nov 18, 2016

### BvU

IF the function is even ($\ f(x) = f(-x)\$) then yes.
You can give it a try...

17. Nov 18, 2016

### core1985

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

18. Nov 18, 2016

### core1985

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

19. Nov 18, 2016

### core1985

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

20. Nov 18, 2016

### BvU

Yes you can
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...