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In mathematics, the definite integral ()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.
Then | | = (()) +, where sgn(x) is the sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive. This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus and philosopher Democritus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. [1]
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 following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
For negative values of n (negative powers of x), there is a singularity at x = 0, and thus the definite integral is based at 1, rather than 0, yielding: = + (+) Further, for negative fractional (non-integer) values of n, the power x n is not well-defined, hence the indefinite integral is only defined for positive x.
Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function. [1] However, it is conventional to omit this from the notation.