Integration by parts and improper integral

[David Fishber]

I would like to solve the following integral but I am unsure of the best way to solve it:
$\int_{0}^{H}xsin(\frac{w}{x})cos(\frac{x}{w})cosh(\frac{H}{w})dx$

Is it possible to use integration by parts??

Thanks in advance

 

[homology]

Note 1: this is an improper integral so even if we find antiderivatives we'll need to figure out if the integral exists. Although a couple quick graphs seem to indicate that the discontinuity is removable so we're probably okay.

Note 2: Well the cosh term doesn't matter (assuming w is independent of x) so that's the first simplification. I took a stab at it letting dv=cos(x/w) and u=xsin(w/x) and obtained:

$$wx\sin(\frac{x}{w})\sin(\frac{w}{x})|_0^H-\int_0^H w\sin(\frac{x}{w})\sin(\frac{w}{x})-\frac{w^2}{x}\sin(\frac{x}{w})\cos(\frac{w}{x})$$

I don't think this is going to go anywhere however. Do you need an exact value or just some approximate values? Are there any constraints on H?

 

[paulfr]

I would set H=w=1 and solve numerically
Then repeat for 2 and so on.
I do not think an AntiDerivative can be found for this integrand, but I am not sure about that.

 

[lavinia]

I would like to solve the following integral but I am unsure of the best way to solve it:
$\int_{0}^{H}xsin(\frac{w}{x})cos(\frac{x}{w})cosh(\frac{H}{w})dx$

Is it possible to use integration by parts??

Thanks in advance

$cosh(\frac{H}{w})$

is a constant.

But the remaining term probably does not have a closed form solution.