$\int \cos \left(\log \frac{x}{a}\right) d x$ is equal to :

$\frac{x}{2}\left[\cos \left(\ln \frac{x}{a}\right)+\sin \left(\ln \frac{x}{a}\right)\right]+c$

Let $I=\int \cos \left(\ln \frac{x}{a}\right) d x$

Let $\ln \left(\frac{x}{a}\right)=t \Rightarrow x=a . e^{t}$

∴ $dx=a e^{t} dt$

$=a \int e^t \cos t d t=\frac{a e^t}{2}[\cos t+\sin t]+c$

$=\frac{x}{2}\left[\cos \left(\ln \frac{x}{a}\right)+\sin \left(\ln \frac{x}{a}\right)\right]+c$

Hence (2) is the correct answer.