Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / D / D4 / 1 / a / Solution

The $k$th derivative of the integrand is $(\cos x{)}^k\cos (t\cos x+k\pi /2)$, whose absolute value is at most one. The integration interval has finite measure. Repeated application of the differentiation theorem therefore givesDominated convergence also gives continuity of every derivative, proving infinite differentiability.