Solution
= 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 gives
$$
\boxed{J_0^{(k)}(t)=\frac2\pi\int_0^{\pi/2}(\cos x)^k\cos(t\cos x+k\pi/2)\,dx\quad(k\ge0).}
$$
<Dominated convergence> also gives <continuity> of every <derivative>, proving infinite <differentiability>.