# Integral Trouble

While doing some quantum mechanics homework I came accross 3 integrals that are giving me trouble.

1) dx/(a^2+x^2)^2 from negative infinity to infinity

2) (x^2 dx)/(a^2+x^2)^2 from negative infinity to infinity

3) (sinkx)^2/x^2 from 0 to infinity

I would appreciate any help that anyone has to offer. #3 resembles the Dirichlet Integral (if we drop off the squares) and then it would be pi/2. All k's and a's are just constants.

Thanks,

jcain6

jcain6 said:
While doing some quantum mechanics homework I came accross 3 integrals that are giving me trouble.

1) dx/(a^2+x^2)^2 from negative infinity to infinity

2) (x^2 dx)/(a^2+x^2)^2 from negative infinity to infinity

3) (sinkx)^2/x^2 from 0 to infinity

I would appreciate any help that anyone has to offer. #3 resembles the Dirichlet Integral (if we drop off the squares) and then it would be pi/2. All k's and a's are just constants.

Thanks,

jcain6

For the first one, try $x=a\tan{\theta}$. That will be really messy though, there must be a better way...

lurflurf
Homework Helper
For (2) use "integration by parts"
$$\int_a^b u \ dv=uv|_a^b-\int_a^b v \ du$$
chose u=x dv=x/(x^2+a^2)^2
for (1) note
a^2(1)+(2)=pi/a
for (1) and (2) I assume a>0
for (3)
differentiate w/ respect k to obtain a Dirichlet Integral
note (3)=0 if k=0 then integrate w/respect k to find (3)

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I suggest you try to use some complex analysis. It seemed to work for me, on the first one anyway.

Spinny said:
I suggest you try to use some complex analysis. It seemed to work for me, on the first one anyway.

I'm curious. How did you do this?

lurflurf
Homework Helper
apmcavoy said:
I'm curious. How did you do this?
Residue Theorem
$$\oint_C f(z) \ dz=2\pi i\sum res(f(z))_{z=a}$$
In words the Integral of a function around a closed contour equals the sum of the residues of the sigularities of the function times 2pi*i. For (1) and (2) the functions themselves with a contour of a large semicircle in the upper half plan works fine. The sine one is a bit tricker use f(z)=1-exp(2kiz) and a contour of semicircle in upperhalf plane with indentation at origin. There is however no need to use the residue theorem as I above noted the easy way to do them.