1. **State the problem:** Compute the definite integral $$\int_0^2 e^{-5x} \, dx$$ using exact values. 2. **Recall the formula:** The integral of an exponential function $$e^{ax}$$ is $$\frac{1}{a} e^{ax} + C$$ where $$a \neq 0$$. 3. **Apply the formula:** Here, $$a = -5$$, so $$\int e^{-5x} \, dx = \frac{1}{-5} e^{-5x} + C = -\frac{1}{5} e^{-5x} + C$$ 4. **Evaluate the definite integral:** $$\int_0^2 e^{-5x} \, dx = \left[-\frac{1}{5} e^{-5x}\right]_0^2 = -\frac{1}{5} e^{-10} + \frac{1}{5} e^{0}$$ 5. **Simplify:** $$= \frac{1}{5} (1 - e^{-10})$$ 6. **Final answer:** $$\boxed{\frac{1 - e^{-10}}{5}}$$ This is the exact value of the integral.