Answer
$\;\;x^{3i}=\cos (3\ln x)+i\sin (3\ln x)$
Work Step by Step
We will use the Euler's formula
\[e^{i\theta}=\cos \theta+i\sin\theta\]
$x^{3i}=e^{3i\ln x}$
$x^{3i}=e^{i(3\ln x)}$
By using Euler's Formula
$x^{3i}=\cos (3\ln x)+i\sin (3\ln x)$
Here,
$\;\;u(x)=\cos (3\ln x)$ and $\;\;v(x)=\sin (3\ln x)$
Hence $\;\;x^{3i}=\cos (3\ln x)+i\sin (3\ln x)$.
