g_n integrable on R

Question

Let g_n (x) = 1 if x=0

           sin x /x  if -n<= x <= n
           0         if x<-n  or  x>n 

show that for every n, g_n is integrable (Lebesgue integrable) on R.

Answer 1

Without the R-L Theorem:

$$\int\limits_{-\infty}^\infty g_n(x)dx\stackrel ?=\int\limits_{-\infty}^{-n}g_n(x)dx+\int\limits_{-n}^ng_n(x)dx+\int\limits_n^\infty g_n(x)dx$$

Since each and every one of the three integrals on the RHS above exists, you can delete now that question mark (?) and we're done.