Question

find the exact area of the region bounded by the curve y=1+e^(2x-1), the x-axis and the lines x=1/2 and x =2

Answer 1

unless you already have it in your head, always worth scribbling down a rough plot. eg to check for weird stuff in the interval.
but you need to do this
\[\int\limits_{1/2}^{2} 1+e^{2x-1} \ dx\]
you should get
e^3 / 2 + 1