Solve Integral with Fourier Transform - Get Help Now!

[doublemint]

Hello!
Can someone help me with this.
Evaluate:
the integral from zero to infinite of ((xcos(x)-sin(x))/x^3)cos(x/2)dx

I think it has to do with Fouriers Transform but I am just stuck.
Any help would be appreciated!
Thank You

 

[fzero]

Make a substitution $$x = 2u$$. Use the double angle formulas to expand $$\sin 2u$$ and $$\cos 2u$$. The remaining integrals can be evaluated in the same manner as the Dirichlet integral: http://en.wikipedia.org/wiki/Dirichlet_integral

 

[doublemint]

Thanks for the response fzero. There is one problem I have, my class has not covered the Dirichlet Integrals..Is there another method of solving this integral?

 

[fzero]

Thanks for the response fzero. There is one problem I have, my class has not covered the Dirichlet Integrals..Is there another method of solving this integral?

It doesn't matter whether you covered them or not. I linked you to that page because it shows you how to evaluate them. Adapt the techniques to the integrals you have to do or find another way to do them.

 

[jackmell]

Thanks for the response fzero. There is one problem I have, my class has not covered the Dirichlet Integrals..Is there another method of solving this integral?

You could formally take the Fourier Transform. Let:

$$\mathcal{F}\left\{f(x)\right\}=\frac{1}{\sqrt{2 \pi }}\int _{-\infty }^{\infty }f(t)e^{i\omega t}dt$$and suppose that we are given:

$$\mathcal{F}\left\{\frac{x\cos(x)-\sin(x)}{x^3}\right\}=\frac{1}{4} \sqrt{\frac{\pi }{2}} \left(-1+w^2\right) (\text{Sign}[1-w]+\text{Sign}[1+w])$$

Where Sign(x) is either -1,0 or 1 depending on x being negative, zero, or positive. Can you now solve your integral?