Evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.181. 16∫ (dt / t^4)1 185. (π/4)∫ sec θ t and θ0 **5.3 The Fundamental Theorem of Calculus**The Fundamental Theorem of Calculus, Part 2:If \( f \) is continuous over the interval \([a, b]\) and \( F(x) \) is any antiderivative of \( f(x) \), then\[\int_a^b f(x) \, dx = F(b) - F(a).\]We often see the notation \( F(x) \Big|_a^b \) to denote the expression \( F(b) - F(a) \). We use this vertical bar and associated limits \( a \) and \( b \) to indicate that we should evaluate the function \( F(x) \) at the upper limit (in this case, \( b \)), and subtract the value of the function \( F(x) \) evaluated at the lower limit (in this case, \( a \)).The Fundamental Theorem of Calculus, Part 2 (also known as the evaluation theorem) states that if we can find an antiderivative for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and subtracting.**Proof**Let \( P = \{x_i\}, i = 0, 1, \ldots, n \) be a regular partition of \([a, b]\). Then, we can write\[F(b) - F(a) = F(x_n) - F(x_0)\]\[= [F(x_n) - F(x_{n-1})] + [F(x_{n-1}) - F(x_{n-2})] + \ldots + [F(x_1) - F(x_0)]\]\[= \sum_{i=1}^n [F(x_i) - F(x_{i-1})].\] **5.3 The Fundamental Theorem of Calculus**This section discusses an essential concept in calculus, the Fundamental Theorem of Calculus. The theorem bridges the concept of differentiation and integration. Here’s a detailed explanation:The expression \(\sum_{i=1}^{n} \left[F(x_i) - F(x_{i-1})\right]\) represents the approximate sum of differences in the antiderivative \(F\) over intervals \([x_{i-1}, x_i]\).1. **Antiderivative \(F\):** It is explained that \(F\) is an antiderivative of \(f\) over \([a, b]\). Using the Mean Value Theorem for each interval \([x_{i-1}, x_i]\), there exists a \(c_i\) such that: \[ F(x_i) - F(x_{i-1}) = F'(c_i)(x_i - x_{i-1}) = f(c_i)\Delta x \]2. **Equation Substitution:** By substituting the above result into the sum, the equation becomes: \[ F(b) - F(a) = \sum_{i=1}^{n} f(c_i)\Delta x \]3. **Limit and Integration:** Taking the limit as \(n \to \infty\), the expression: \[ F(b) - F(a) = \lim_{n \to \infty}\sum_{i=1}^{n} f(c_i)\Delta x \] Approximates to the definite integral: \[ F(b) - F(a) = \int_{a}^{b} f(x) \, dx \]This theorem provides the framework for evaluating definite integrals using antiderivatives, simplifying calculations across various applications in calculus.

By Fundamental theorem of calculus .part 2 ∫abf(x)dx=F(b)-F(a) where f(x) is continuous on the [a,b] and F(x) is antiderivative of f(x), that is f(x)=ddxF(x)181 ∫116dtt4 Here f(t)=1t4ddt-13t3=1t4 That is antiderivative of f(t) is F(t)=-13t3 Therefore, by Fundamental Theorem of calculus, ∫116dtt4=F(16)-F(1)=-13(16)3+13(1)3=409512288185 ∫0π4secθtanθdθ Here f(θ)=secθtanθddtsecθ=secθtanθ That is antiderivative of f(θ) is F(θ)=secθ Therefore, by Fundamental Theorem of calculus, ∫0π4secθtanθdθ=Fπ4-F(0)=secπ4-sec(0)=2-1Ans: ∫116dtt4=409512288∫0π4secθtanθdθ=2-1