Answer
$$\frac{1}{2}(1-\cos 1)$$
Work Step by Step
Given $$y=\sin x, \quad y=x \sin \left(x^{2}\right), \quad 0 \leq x \leq 1$$
Since
$$\sin x\geq x \sin \left(x^{2}\right), \quad 0 \leq x \leq 1 $$
Then
\begin{aligned}
A &=\int_{0}^{1}\left(\sin x-x \sin \left(x^{2}\right)\right) d x\\
&=\int_{0}^{1} \sin x d x-\int_{0}^{1} x \sin \left(x^{2}\right) d x \\
&=- \cos x+\frac{1}{2}\cos(x^2)\bigg|_{0} ^{1}\\
&=\frac{1}{2}(1-\cos 1)
\end{aligned}
