Answer
$$\frac{1}{4}\left[x^{4} \sin x^{4}+\cos x^{4}\right]+C$$
Work Step by Step
Given $$ \int x^{7} \cos \left(x^{4}\right) d x$$
Let $$ z=x^4 \ \ \ \ \ \ dz=4x^3 dx $$
Then
$$ \int x^{7} \cos \left(x^{4}\right) d x= \frac{1}{4} \int z \cos (z) d z$$
Use integration by parts
\begin{align*}
u&= z\ \ \ \ \ \ \ \ \ dv=\cos z\\
du&=dz\ \ \ \ \ \ \ \ \ v=\sin z
\end{align*}
Hence
\begin{aligned}
\int x^{7} \cos \left(x^{4}\right) d x &=\frac{1}{4} \int z \cos (z) d z \\
&=\frac{1}{4}\left[z \sin z-\int \sin z d z\right] \\
&=\frac{1}{4}[z \sin z+\cos z]+C \\
&=\frac{1}{4}\left[x^{4} \sin x^{4}+\cos x^{4}\right]+C
\end{aligned}
