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

Question

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?

Answer 1

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