In this exercise we will establish the important formula: (a) Let G(t) := «10 [e-t2(x2+1)/(x2 + 1)]dx for t > 0. Since the integrand is

In this exercise we will establish the important formula:

(a) Let G(t) := ˆ«10 [e-t2(x2+1)/(x2 + 1)]dx for t > 0. Since the integrand is dominated by 1/(x2 + 1) for t > 0, then G is continuous on [0, ˆž). Moreover, G(0) = Arctan 1 = 1/4 Ï€ and it follows from the Dominated Convergence Theorem that G(t) †’ 0 as t †’ ˆž.
(b) The partial derivative of the integrand with respect to t is bounded for t > 0, x ˆˆ [0, 1], so Gʹ(t) = -2te-t2 ˆ«10 e-t2x2 dx = -2e-t2 ˆ«t0 e-u2 du.
(c) If we set F(t) := [ˆ«t0 e-x2 dx]2, then the Fundamental Theorem 10.1.11 yields Fʹ(t) = 2e-t2
ˆ«t0 e-x2 dx for t > 0, from which Fʹ(t) + Gʹ(t) = 0 for all t > 0. Therefore, F(t) + G(t) = C for all t > 0.
(d) Using F(0) = 0, G(0) = 1/4 Ï€ and limt†’ˆžG(t) = 0, we conclude that limt†’ˆžF(t) = 1/4 Ï€, so that formula (17) holds.

(17)