Homogeneous DE of the first order

I must solve $\displaystyle \dot x + x\cos t=0$.

I realize it's separable, and solving it I reach $\displaystyle x=e^{-\sin (t) + C}$ where $\displaystyle C$ is a constant.

However, solving it via the integrating factor method, I do not reach the same result.

The IF is $\displaystyle e^{\sin t}$.

Multiplying both sides by it, I get $\displaystyle \frac{dx}{dt} e^{\sin t}+x\cos (t) e^{\sin t}=0$.

Integrating with respect to t, I get $\displaystyle e^{\sin t}x=0$.

Now clearly $\displaystyle x\equiv 0$.

What am I doing wrong?