Answer
$$\int\frac{\sin(\ln x)}{x}dx=-\cos(\ln x)+c$$
Work Step by Step
To evaluate the integral
$$\int\frac{\sin(\ln x)}{x}dx$$
we will use substitution $\ln x=t$ which gives us $\frac{dx}{x}=dt$. Putting this into the integral we get:
$$\int\frac{\sin(\ln x)}{x}dx=\int\sin tdx=-\cos t+c$$
where $c$ is arbitrary constant. Now we have to express solution in terms of $x$:
$$\int\frac{\sin(\ln x)}{x}dx=-\cos t+c=-\cos(\ln x)+c$$
