Answer
$$f\left( x \right) = 2{e^{2x}}{\text{ and }}a = \ln 2$$
Work Step by Step
$$\eqalign{
& 4 + \int_a^x {f\left( t \right)dt} = {e^{2x}} \cr
& {\text{Subtract 4 from both sides}} \cr
& \int_a^x {f\left( t \right)dt} = {e^{2x}} - 4 \cr
& {\text{Differentiate both sides with respect to }}x \cr
& \frac{d}{{dx}}\left[ {\int_a^x {f\left( t \right)dt} } \right] = \frac{d}{{dx}}\left[ {{e^{2x}} - 4} \right] \cr
& f\left( x \right) = 2{e^{2x}} \cr
& {\text{Let }}f\left( t \right) = 2{e^{2t}},{\text{ then}} \cr
& 4 + \int_a^x {2{e^{2t}}dt} = {e^{2x}} \cr
& 4 + \left[ {{e^{2t}}} \right]_a^x = {e^{2x}} \cr
& \left[ {{e^{2t}}} \right]_a^x = {e^{2x}} - 4 \cr
& {e^{2x}} - {e^{2a}} = {e^{2x}} - 4 \cr
& {\text{Solve for }}a \cr
& {e^{2a}} = 4 \cr
& \ln {e^{2a}} = \ln 4 \cr
& 2a = \ln 4 \cr
& a = \frac{1}{2}\ln 4 \cr
& a = \ln 2 \cr
& \cr
& {\text{Then, }} \cr
& f\left( x \right) = 2{e^{2x}}{\text{ and }}a = \ln 2 \cr} $$
