How can I calculate the integral:
$$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx$$ ??
I got stuck.. :/
Could you give me some hint??
Do I have to use the following formula??
$\displaystyle{\sin{(A)} \sin{(B)}=\frac{\cos{(A-B)}-\cos{(A+B)}}{2}}$
[Math] $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
calculusdefinite integralsimproper-integralsintegration
Best Answer
This integral does not converge but one can make some sense of it anyway as a generalized function of $a$ and $b$. Specifically, the result is a sum of Dirac delta functions. To show this, use the Euler identity to express the sine functions as exponential functions. After a little simplification, you will have several terms and each can be evaluated using the Dirac delta function's Fourier expansion: \begin{equation} \delta(p) = {1\over2\pi}\int_{-\infty}^\infty dx e^{i p x} \end{equation}
Edit: To be clear, this is a somewhat advanced analysis. If this was a homework problem for an elementary class and you've never heard of any of the techniques or formulas I've mentioned, the sought-after answer is probably simply "integral does not exist" which can be demonstrated via the arguments made by other posters.