It is known that the area under the curve $~y=e^x~$ from $~x=0~$ to $~x=1~$ is $~e-1\,$; therefore we know that $\qquad \displaystyle \int_0^1e^x\,dx=e-1$ and is equal to the area of the shaded region shown below. $\qquad$ With this knowledge, evaluate each of the following. $~~~$(a)$~~~ \displaystyle \int_1^1\big(e^x+2\big)\,dx=$ [[☃ expression 1]] $~~~$(b)$~~~ \displaystyle \int_2^3e^{x-2}\,dx=$ [[☃ expression 2]] $~~~$(c)$~~~ \displaystyle \int_0^1\big(3-e^x\big)\,dx=$ [[☃ expression 3]]