Embrace it

Although it is not my original answer,problem can be approached by contour integral.( I just pointed out here) Let sinx= im(e^ix) .now integration become= integration of e^iz/z in contour of a large half circle -R to R and a small circle -e to e------integration has now four parts I1----integral to Closed R I2------ -R to -e I3--- integral to Closed e I4 ---- e to R = 0 as closed integral. Doing all these integral we can find the result. Abhishek this problem was solved by mukesh sir.

we can easily reach to the answer by just taking the laplace transformation of sin x / x --> L(sin x/x) = (integral) (0 to infinity) )((sinx/x)*exp(-sx)dx) which will be equal to (integral) (s to inf) (1/(1+s^2)) ie (Atans)(s to inf) which will lead to pi/2-(Atans) . by observing that if s=0 in basic formula of laplace the integral will reduce to the ques itself hence pi/2-0 would be the answer