Solution
= Solution
The <chain rule> identifies an <antiderivative> directly:
$$
\frac{d}{dx}e^{\sin(x^2)}=2x\cos(x^2)e^{\sin(x^2)}.
$$
The upper limit in the PDF is $\sqrt{\pi/2}$. By the <fundamental theorem of calculus>, the required <integral> is
$$
\boxed{\frac12\left[e^{\sin(x^2)}\right]_0^{\sqrt{\pi/2}}=\frac{e-1}{2}}.
$$