Question

How would you integrate the\[\int_{0}^{\infty} \frac{\sin x}{x}dx\]

Answer 1

So you're looking for the numerical answer, I guess? pi/2

Answer 2

because my first thought was to use series representation.

Answer 3

yeah...i wanna see different methods

Answer 4

hmm yes, I believe I saw you using Dirichlet method once already.

Answer 5

\[\large \int_0^{\infty}\frac{\sin (t)}{t}\text{d}t=\int_0^{\infty}\sin (t) \left ( \int_0^{\infty}e^{-st}\text{d}s\right ) \text{d}t=\int_0^{\infty}\int_0^{\infty}\sin (t) e^{-st} \text{d}s~\text{d}t\]\[\large =\int_0^{\infty}\mathcal{L}\{\sin (t)\}_{(s)}\text{d}s=\int_0^{\infty}\frac{\text{d}s}{1+s^2}=\frac{\pi}{2}\]

Answer 6

It's a cool method for integrals of the form \(\int_0^{\infty}\frac{f(t)}{t}\text{d}t\):\[\int_0^{\infty}\frac{f(t)}{t}\text{d}t=\int_0^{\infty} f(t)\left ( \int_0^{\infty} e^{-st}\text{d}s\right ) \text{d}t=\int_0^{\infty}\int_0^{\infty}f(t)e^{-st}\text{d}s~\text{d}t\]\[=\int_0^{\infty}\int_0^{\infty}f(t)~e^{-st}\text{d}t~\text{d}s=\int_0^{\infty}\mathcal{L}\{f(t)\}_{(s)}\text{d}s\]

Answer 7

@Herp_Derp

thats exactly what i wanted to say about integrals of that form

making double integrals and using laplace