1. **Stating the problem:** Evaluate the improper integral $$\int_0^\infty 7e^x \, dx$$. 2. **Formula and rules:** The integral of an exponential function $$e^x$$ is $$e^x$$ itself. For improper integrals with infinite limits, we evaluate the limit as the upper bound approaches infinity. 3. **Intermediate work:** $$\int_0^\infty 7e^x \, dx = 7 \int_0^\infty e^x \, dx$$ 4. Evaluate the integral: $$7 \int_0^\infty e^x \, dx = 7 \left[ e^x \right]_0^\infty = 7 \left( \lim_{b \to \infty} e^b - e^0 \right)$$ 5. Since $$\lim_{b \to \infty} e^b = \infty$$, the integral diverges to infinity. **Final answer:** The integral $$\int_0^\infty 7e^x \, dx$$ diverges and does not converge to a finite value.