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It states that if a function is continuous on a closed interval [a, b], and if F(x) is an antiderivative of f(x) on that interval, then the definite integral of f(x) from a to b can be calculated as the difference between the values of F(x) at the endpoints a and b. The theorem is divided into two parts: the first part deals with the fundamental theorem of calculus for definite integrals, while the second part focuses on the fundamental theorem of calculus for indefinite integrals. The first part states that if f(x) is a continuous function on [a, b], then the function g(x) defined as the integral of f(t) with respect to t from a to x is continuous on [a, b] and differentiable on (a, b), with g'(x) equal to f(x) for every x in (a, b). This means that we can find the exact value of the definite integral by evaluating the antiderivative at the endpoints. The second part states that if f(x) is a continuous function on an interval I, and if F(x) is an antiderivative of f(x) on I, then for any x in I, the definite integral of f(x) from a to x is equal to F(x) minus F(a), where a is a fixed point in I. This implies that we can find an antiderivative of a function by evaluating the definite integral. The fundamental theorem of calculus has numerous practical applications in mathematics, physics, engineering, and other scientific disciplines. It provides a powerful tool for solving problems involving integration and differentiation, and it forms the basis for many advanced concepts in calculus, such as the calculation of areas and volumes, the evaluation of improper integrals, and the solution of differential equations.