2.) $$\int_{0}^{1} e^{2x} dx = $$

Let's solve the definite integral: $$\int_{0}^{1} e^{2x} dx$$ We know that the integral of $$e^{ax}$$ is $$\frac{1}{a}e^{ax}$$. Therefore, $$\int e^{2x} dx = \frac{1}{2} e^{2x} + C$$ Now we need to evaluate the definite integral from 0 to 1: $$\int_{0}^{1} e^{2x} dx = \frac{1}{2} e^{2x} \Big|_{0}^{1} = \frac{1}{2} (e^{2(1)} - e^{2(0)}) = \frac{1}{2} (e^2 - e^0) = \frac{1}{2} (e^2 - 1)$$ So, the result is: $$\frac{1}{2}(e^2 - 1)$$ Answer: $$\frac{1}{2}(e^2 - 1)$$