Answer
$$\frac{1}{3}\sin \left(1\right)$$
Work Step by Step
Given
$$\int_{0}^{1}v^2\cos(v^3)dv$$
Let $x=v^3\ \Rightarrow \ dx=3v^2dv $, at $v=0\to x=0$, at $v=1\to x=1$, then
\begin{align*}
\int_{0}^{1}v^2\cos(v^3)dv&=\frac{1}{3}\int_{0}^{1}\cos(x)dx\\
&=\frac{1}{3}\sin x\bigg|_{0}^{1}\\
&=\frac{1}{3}\sin \left(1\right)
\end{align*}
