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Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, [1] is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.
In other words if an element a square root u of an element a exists, then the only square roots of a are u and −u. The only square root of 0 in an integral domain is 0 itself. In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that −u = u.
Many of the following antiderivatives have a term of the form ln |ax + b|.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]
(Note that the value of the expression is independent of the value of n, which is why it does not appear in the integral.) ∫ x x ⋅ ⋅ x ⏟ m d x = ∑ n = 0 m ( − 1 ) n ( n + 1 ) n − 1 n !
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.
Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit. [8] [9] This is useful for electrical engineers in calculating the "AC only" RMS of a signal.
Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure (quadrature or squaring), as in the quadrature of the circle. The term is also sometimes used to describe the numerical solution of differential equations .