1. **Problem statement:** Calculate the definite integrals using the Fundamental Theorem of Calculus. 2. **Formula:** The Fundamental Theorem of Calculus states: $$\int_a^b f(x) \, dx = F(b) - F(a)$$ where $F$ is any antiderivative of $f$, i.e., $F' = f$. 3. **Integral a) $$\int_0^5 e^x \, dx$$** - Antiderivative of $e^x$ is $e^x$. - Evaluate: $$F(5) - F(0) = e^5 - e^0 = e^5 - 1$$ 4. **Integral b) $$\int_0^1 (2x - e^x) \, dx$$** - Antiderivative of $2x$ is $x^2$. - Antiderivative of $-e^x$ is $-e^x$. - So, $$F(x) = x^2 - e^x$$ - Evaluate: $$F(1) - F(0) = (1^2 - e^1) - (0^2 - e^0) = (1 - e) - (0 - 1) = 1 - e + 1 = 2 - e$$ 5. **Integral c) $$\int_0^{\ln(2)} (e^x + 1) \, dx$$** - Antiderivative of $e^x$ is $e^x$. - Antiderivative of $1$ is $x$. - So, $$F(x) = e^x + x$$ - Evaluate: $$F(\ln(2)) - F(0) = (e^{\ln(2)} + \ln(2)) - (e^0 + 0) = (2 + \ln(2)) - (1 + 0) = 1 + \ln(2)$$ **Final answers:** - a) $$e^5 - 1$$ - b) $$2 - e$$ - c) $$1 + \ln(2)$$