Integration inequality, why can we pull e^t out of the integral and leave its e?

In the following, at line 3 $$e^t \sin(t)$$ is pulled out of the $$|\cdot|$$ and left as a constant. How does one justify this step?

Since $$x\in [0,1]$$ and since the exponential function is increasing, it follows that $$e^{x}\leq e^{1}=e$$.