Six Rules for Definite Integrals and Bounds on eᵇ

Write the sum rule and the constant-multiple rule as formulas, note that $c$ must not depend on $x$, and record the integral sign as a stretched S for sum.

State $\int_a^a f(x)\,dx = 0$ and the definition $\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$, check both against $F(b) - F(a)$, and write additivity over $[a, b]$ and $[b, c]$ for points in any order.

State that $f(x) \le g(x)$ gives $\int_a^b f(x)\,dx \le \int_a^b g(x)\,dx$ only when $a < b$, record the slower-traveller interpretation, and note that the signs of $f$ and $g$ do not matter.

Integrate $e^x \ge 1$ from $0$ to $b$ to reach $e^b \ge 1 + b$, integrate that from $0$ to $b$ to reach $e^b \ge 1 + b + \frac{b^2}{2}$, and record where $b \ge 0$ is needed.