Antiderivatives and indefinite integrals from "summary" of Differential Calculus by S Balachandra Rao
Antiderivatives and indefinite integrals are fundamental concepts in calculus that are closely related. An antiderivative of a function f is a function F whose derivative is equal to f. In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x). Antiderivatives are also known as primitives, and finding antiderivatives is an important skill in calculus. When we talk about antiderivatives, we are essentially looking for functions that, when differentiated, give us the original function we started with. This process can be thought of as "undoing" the process of differentiation. Just like how addition and subtraction are inverse operations, differentiation and antidifferentiation are also inverse operations. Indefinite integrals are closely related to antiderivatives. An indefinite integral represents a set of functions that are antiderivatives of a given function. The symbol ∫ f(x) dx represents the indefinite integral of f(x) with respect to x. This symbol is read as "the integral of f(x) with respect to x." Finding the antiderivative of a function f involves finding a function F such that F'(x) = f(x). This process is known as antidifferentiation or integration. The symbol ∫ f(x) dx represents the antiderivative of f(x) with respect to x. The constant of integration, denoted by +C, is added when finding the antiderivative to account for all possible antiderivatives. Indefinite integrals are essentially antiderivatives without definite limits of integration. When we find the antiderivative of a function, we are essentially finding a whole family of functions that differ by a constant. This constant is accounted for by the constant of integration. In summary, antiderivatives and indefinite integrals are important concepts in calculus that involve finding functions whose derivatives match a given function. Antiderivatives are related to indefinite integrals, which represent a set of functions that are antiderivatives of a given function. Antiderivatives are found through the process of antidifferentiation, and the constant of integration accounts for the family of functions that differ by a constant.
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