Given that \(\displaystyle\displaystyle \displaystyle\int_{0}^{\ln 2} \mathrm{f}(x) \, \mathrm{d}x = 8\), evaluate \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{1}{2}} \mathrm{f}(x) + e^{2x} \, \mathrm{d}x + \displaystyle\displaystyle \displaystyle\int_{\frac{1}{2}}^{\ln 2} \mathrm{f}(x) + e^{x} \, \mathrm{d}x\) exactly.
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