Answer: \(\dfrac34+\mathrm e^4+\dfrac{\mathrm e^8}{4}\).

We start with the main method. Simplify the integrand before integrating, then substitute the upper and lower limits carefully.

First expand the integrand:

\((1+\mathrm e^{2x})^2=1+2\mathrm e^{2x}+\mathrm e^{4x}.\)

So

\(\int_0^2(1+\mathrm e^{2x})^2\,dx=\int_0^2\left(1+2\mathrm e^{2x}+\mathrm e^{4x}\right)\,dx.\)

Integrate term by term:

\(\int\left(1+2\mathrm e^{2x}+\mathrm e^{4x}\right)\,dx =x+\mathrm e^{2x}+\frac{\mathrm e^{4x}}{4}.\)

Now apply the limits:

\(\left[x+\mathrm e^{2x}+\frac{\mathrm e^{4x}}{4}\right]_0^2 =\left(2+\mathrm e^4+\frac{\mathrm e^8}{4}\right)-\left(0+1+\frac14\right).\)

Therefore the exact value is

\(\frac34+\mathrm e^4+\frac{\mathrm e^8}{4}.\)

This completes the solution and gives the required result.