DEFINITE integral of sinaxsinbx FROM 0 to infinity

[iqjump123]

Hello, first time poster.

I am putting this question in the PDE section because this was a question I came up while solving a PDE question.

Also, I figured that since this is not a straight homework question, I can post in this category. Mods, feel free to move this post to wherever you see fit. thanks.

I have looked everywhere, including the gredsteyin book of integrals and transforms to find this solution, but I wasn't able to find it. I am especially talking about the one from 0 to infinity as the bounds.

using mathcad just gives me a solution with infinity in it as a variable.

My professor will not tell me what it is, pretty much telling me that I am a fool for not being able to find it. @#%@#%@#%@

Also, another one that I couldn't find it for was:

integral of sin(ax)sin(ax) from 0 to an arbitrary number t?
Any help on this will be appreciated!
Thanks so much.

iqjump123

 

[LCKurtz]

Hello, first time poster.

I am putting this question in the PDE section because this was a question I came up while solving a PDE question.

Also, I figured that since this is not a straight homework question, I can post in this category. Mods, feel free to move this post to wherever you see fit. thanks.

I have looked everywhere, including the gredsteyin book of integrals and transforms to find this solution, but I wasn't able to find it. I am especially talking about the one from 0 to infinity as the bounds.

using mathcad just gives me a solution with infinity in it as a variable.

My professor will not tell me what it is, pretty much telling me that I am a fool for not being able to find it. @#%@#%@#%@

Also, another one that I couldn't find it for was:

integral of sin(ax)sin(ax) from 0 to an arbitrary number t?
Any help on this will be appreciated!
Thanks so much.

iqjump123

Look at the product formulas here if a and b aren't equal:

http://www.sosmath.com/trig/prodform/prodform.html

If a = b you can use the double angle formula for sin²(θ). I don't think you will find the integral converges for t → ∞.

 

[iqjump123]

Look at the product formulas here if a and b aren't equal:

http://www.sosmath.com/trig/prodform/prodform.html

If a = b you can use the double angle formula for sin²(θ). I don't think you will find the integral converges for t → ∞.

Thanks for the info!

I figured as such for the ones with the different values for a and b..
I think I will just write out the indefinite forms and write out the limits manually instead of evaluating it.

If anybody else can shed light on this topic, that will be great!
Thanks very much.

 

[HallsofIvy]

You might remember that cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)
Changing the sign on b: cos(a- b)= cos(a)cos(b)+ sin(a)sin(b)
(because cos(-b)= cos(b) and sin(-b)= -sin(b).

Subtracting the first equation from the second, the "cos(a)cos(b)" terms cancel and we have
cos(a-b)- cos(a+b)= 2sin(a)sin(b) so that

sin(a)sin(b)= (1/2)(cos(a-b)- cos(a+ b)[

and, therefore,
sin(ax)sin(bx)= (1/2)(cos((a-b)x)- cos((a+b)x)

From that, it is easy to get
$$\int_0^A sin(ax)sin(bx)= (1/2)\int_0^A cos((a-b)x)dx- (1/2)\int_0^A cos((a+b)x)$$
$$\frac{1}{2(a-b)}sin((a-b)A- \frac{1}{2(a+b)}sin((a+b)A)$$

But the problem is taking the limit as A goes to infinity!

 

[blue_raver22]

Hello iqjump123,

Thank you for your question. The definite integral of sin(ax)sin(bx) from 0 to infinity is a challenging one to solve. It requires advanced techniques such as contour integration or Fourier transforms.

As for the integral from 0 to an arbitrary number t, you can use the trigonometric identity sin^2(x) = 1/2 - 1/2cos(2x) to rewrite the integral as:

integral of (1/2 - 1/2cos(2ax))dx from 0 to t

= 1/2x - 1/4sin(2ax) from 0 to t

= 1/2t - 1/4sin(2at) - 1/4sin(0)

= 1/2t - 1/4sin(2at)

I hope this helps. Keep exploring and learning, and don't be discouraged by difficult problems. That's how we grow as scientists. Best of luck!