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In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy -plane bounded by the graph of f , the x -axis, and the lines x = a and x = b , such that area above the x -axis adds to the total, and that below the x -axis subtracts from the total.
A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals; Math Major: A Table of Integrals; O'Brien, Francis J. Jr. "500 Integrals of Elementary and Special Functions". Derived integrals of exponential, logarithmic functions and special functions.
Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.
The former expression is written as a definite integral and the latter is written as an indefinite integral. Applying the appropriate limits to the latter expression should yield the former, but the latter is not necessarily equivalent to the former. Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.
Trigonometric identities may help simplify the answer. [1] [2] Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field.
The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:
A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer, we need to take a limit as Δx approaches zero. [46]: 512–522 The symbol of integration is , an elongated S chosen to suggest summation. [46]: 529 The definite integral is written as: