Answer
$$ x^2\sin(x^2)+ \cos(x^2) +C$$
Work Step by Step
Given
$$\int 2 x^{3} \cos x^{2} d x $$
Let
$$z =x^2\ \ \to\ \ dz =2xdx$$
Then
\begin{aligned}\int 2 x^{3} \cos x^{2} d x &= \int z\cos zd z \end{aligned}
Use integration by parts , let
\begin{aligned}
u&= z \ \ \ \ \ \ &dv&= \cos z dz\\
du&= dz \ \ \ \ \ \ &v&= \sin z
\end{aligned}
then
\begin{aligned}
\int 2 x^{3} \cos x^{2} d x &= \int z\cos zd z \\
&= \left( z\sin z- \int \sin zdz\right)\\
&= z\sin z+ \cos z+C\\
&= x^2\sin(x^2)+ \cos(x^2) +C
\end{aligned}
