**Solution.**We want to determine the integral of \sin(\ln(x)):

\begin{align*} I = \int \sin(\ln(x)) dx. \end{align*}

\begin{align*} \int UdV = UV - \int VdU. \end{align*}

\begin{align*} U = \sin(\ln(x)), \quad &dV = dx\\ dU = \frac{\cos(\ln(x))}{x}dx, \quad &V = x. \end{align*}

\begin{align*} \int \sin(\ln(x)) dx = x\sin(\ln(x)) - \int \cos(\ln(x))dx. \end{align*}

\begin{align*} U = \cos(\ln(x)), \quad &dV = dx\\ dU = \frac{-\sin(\ln(x))}{x}dx, \quad &V = x. \end{align*}

\begin{align*} \int \sin(\ln(x)) dx &= x\sin(\ln(x)) - \int \cos(\ln(x))dx \\ &= x\sin(\ln(x)) - (x\cos(\ln(x)) + \int \sin(\ln(x))dx) \\ &= x\sin(\ln(x)) - x\cos(\ln(x)) - \int \sin(\ln(x))dx \\ &= x\sin(\ln(x)) - x\cos(\ln(x)) - I. \end{align*}

\begin{align*} \int \sin(\ln(x)) dx = \frac{1}{2}x(\sin(\ln(x)) - \cos(\ln(x))) + C. \end{align*}