Fundamental Theorem of Calculus The Fundamental Theorem of Calculus establishes a direct connection between the limits of a function and the values of the f...
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus establishes a direct connection between the limits of a function and the values of the function at those points. It provides a rigorous framework for evaluating definite integrals by demonstrating that they are equal to the areas of corresponding rectangles.
Properties of the Fundamental Theorem of Calculus
Additivity: If (f(x)) and (g(x)) are functions, then (f(x) + g(x)) is also integrable and (∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx).
Constant Multiple Rule: If (c) is a constant, then (c\cdot f(x)) is also integrable and (∫(c\cdot f(x))dx = c∫f(x)dx).
Integration by Parts: If (u(x)) and (v(x)) are functions, then their integration by parts is given by the formula (\int u(x)v(x)dx = uv(x) - ∫ v(x)u(x)dx).
Chain Rule for Integrals: If (f(x) = g(x)^n), then (∫f(x)dx = n∫g(x)^{n-1}dx).
These properties allow us to manipulate integrals and evaluate them directly using basic algebraic and geometric concepts