Question

is it possible integrate (sinx/x) from 0 to infinity
without using engineering math????

Answer 1

\[\int\limits_{0}^{\infty} \frac{\sin x}{x} dx\]
?

Answer 2

yes

Answer 3

are u familiar with laplace transform?

Answer 4

yes !!

Answer 5

\[\int\limits_{0}^{\infty} \frac{\sin x}{x} dx= \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} e^{-xy} dy \sin x \ dx=\int\limits_{0}^{\infty} \int\limits_{0}^{\infty} e^{-xy} \sin x \ dy dx=\int\limits_{0}^{\infty} \int\limits_{0}^{\infty} e^{-xy} \sin x \ dx dy\]
intervals are equal so dx dy=dy dx
now u have (using laplace transform)
\[\int\limits_{0}^{\infty} \int\limits_{0}^{\infty} e^{-xy} \sin x \ dx dy=\int\limits_{0}^{\infty} \frac{1}{1+y^2} dy=tan^{-1} \infty=\frac{\pi}{2} \]

Answer 6

oh very well thank u

Answer 7

welcome

Answer 8

@mukushla