If t > 0, define E(t) := «0 [(e-txsin x)/x]dx . (a) Show that E exists and is continuous for t > a > 0.

If t > 0, define E(t) := ˆ«ˆž0 [(e-txsin x)/x]dx .
(a) Show that E exists and is continuous for t > a > 0. Moreover, E(t) †’ 0 as t †’ ˆž.

(c) Deduce that E(t) = 1/2 π - Arctan t for t > 0.
(d) Explain why we cannot use the formula in (c) to obtain equation (12).

e-tsin x at Se-