1. The problem is to complete the blanks in the explanation of Integration by Substitution for Definite Integrals. 2. From the given formula: $$\int_a^b f(g(t)) g'(t) dt = [F(g(t))]_a^b = F(g(\beta)) - F(\_ )$$ We know that the limits change according to the substitution. 3. Since the substitution is $x = g(t)$, when $t = a$, $x = g(a)$ and when $t = \beta$, $x = g(\beta)$. 4. Therefore, the missing term in the blank is $g(a)$ because the lower limit $a$ in $x$ corresponds to $g(a)$. 5. So the expression becomes: $$F(g(\beta)) - F(g(a)) = F(b) - F(a) = [F(x)]_a^b$$ 6. This confirms the substitution rule for definite integrals: $$\int_a^b f(x) dx = \int_a^{\beta} f(g(t)) g'(t) dt$$ where $a = g(a)$ and $b = g(\beta)$. Final answers for the blanks: - The first blank is $g(a)$ - The second blank is $a$ Hence, the completed expression is: $$\int_a^b f(g(t)) g'(t) dt = [F(g(t))]_a^b = F(g(\beta)) - F(g(a)) = F(b) - F(a) = [F(x)]_a^b$$