Antiderivative

Antiderivative

An Overview of Riemann Integration

Published in James K. Peterson, Basic Analysis IV: Measure Theory and Integration, 2020

The common symbol for the antiderivative of f has evolved to be ∫ f because of the close connection between the antiderivative of f and the Riemann integral of f which is given in the Cauchy Fundamental Theorem of Calculus. The usual Riemann integral, ∫ba f(t) dt of f on [a, b] computes a definite value – hence, the symbol ∫ba f(t) dt is usually referred to as the definite integral of f on [a, b] to contrast it with the family of functions represented by the antiderivative ∫ f. Since the antiderivatives are arbitrary up to a constant, most of us refer to the antiderivative as the indefinite integral of f. Also, we hardly ever say “let’s find the antiderivative of f” – instead, we just say, “let’s integrate f”. We will begin using this shorthand now!

RIP: row integration by parts

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2022

J. A. Rock

Integration by parts is a classic result in calculus that provides a type of product rule for finding antiderivatives or integrals of products. A goal of this paper is to convince readers that by focusing on the rows – hence the name Row Integration by Parts (RIP) – a tabular method can provide a streamlined approach for any application of integration by parts. RIP is an efficient bookkeeping tool with no loss of information. The key is that each row represents an integral and tables are to be constructed one row at a time.

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An Overview of Riemann Integration

RIP: row integration by parts