1. **State the problem:** We need to evaluate the integral $$\int \cos(\ln x) \, dx$$. 2. **Recall the formula and substitution:** Let us use the substitution $$t = \ln x$$, so that $$x = e^t$$ and $$dx = e^t dt$$. 3. **Rewrite the integral:** Substituting, we get $$\int \cos(\ln x) \, dx = \int \cos(t) e^t \, dt$$. 4. **Integrate by parts:** To integrate $$\int e^t \cos t \, dt$$, use integration by parts or recall the formula: $$\int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2 + b^2} (a \cos(bt) + b \sin(bt)) + C$$. Here, $$a=1$$ and $$b=1$$, so $$\int e^t \cos t \, dt = \frac{e^t}{1^2 + 1^2} (1 \cdot \cos t + 1 \cdot \sin t) + C = \frac{e^t}{2} (\cos t + \sin t) + C$$. 5. **Back-substitute:** Replace $$t = \ln x$$ and $$e^t = x$$: $$\int \cos(\ln x) \, dx = \frac{x}{2} (\cos(\ln x) + \sin(\ln x)) + C$$. **Final answer:** $$\boxed{\int \cos(\ln x) \, dx = \frac{x}{2} (\cos(\ln x) + \sin(\ln x)) + C}$$