Answer
$$\frac{\sinh 2}{2}.$$
Work Step by Step
Let $ u=2t $, then $ du=2dt $, $ t:=0\to 1, u:=0\to 2$.
Now, we have
$$\int_0^1\cosh (2t) dt =\frac{1}{2}\int_0^2\cosh u du \\
=\frac{1}{2} \sinh u|_0^2=\frac{1}{2}(\sinh 2-\sinh 0)= \frac{\sinh 2}{2}.$$
