ordinary differential equations - Is this constant of integration necessary at this step?

I came across a differential equation:

$$\frac{dy}{dx}=\frac{\sin(\log x)}{\log y}$$.
Here is what I tried to do:

I transformed it into this form $$\log y dy=\sin(\log x)dx$$ i.e. $$\int \log y dy=\int \sin(\log x)dx\dots(2)$$ and after that I used integration by parts to finish off the problem.

However,I was told by my teacher that it should instead be $$\int \log y dy=\int \sin(\log x)dx+C$$ where $C$ is a constant of a integration.I argued that the integration had not yet been carried out and so there was no need for the constant.(and I was told it $had$ to be there.)

Can anyone please convince me why my teacher is right and I wrong?

Thanks.