This is the definite integral form; the indefinite integral form is:. There are additional forms , listed below. Together with the linearity of the integral, this formula allows one to compute the integrals of all polynomials. 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.
The improper integral is often extended to negative values of x via the conventional choice:.
The Britannica Guide to Analysis and Calculus (Hardcover)
Note the use of the absolute value in the indefinite integral; this is to provide a unified form for the integral, and means that the integral of this odd function is an even function, though the logarithm is only defined for positive inputs, and in fact, different constant values of C can be chosen on either side of 0, since these do not change the derivative.
The more general form is thus: . As with the indefinite integral, this is special to the real numbers, and does not extend over the complex numbers.
The integral can also be written with indexes shifted, which simplify the result and make the relation to n -dimensional differentiation and the n -cube clearer:. This is shown from the binomial formula and the definition of the derivative — and thus by the fundamental theorem of calculus the antiderivative is the integral. Further, there is an n -fold cyclic symmetry of the n -cube around the diagonal cycling these pyramids for which a pyramid is a fundamental domain.
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In the case of the cube 3-cube , this is how the volume of a pyramid was originally rigorously established: the cube has 3-fold symmetry, with fundamental domain a pyramids, dividing the cube into 3 pyramids, corresponding to the fact that the volume of a pyramid is one third of the base times the height. Speedy Publishing LLC. Happily Never After. Mitchell Symons. Britannica Educational Publishing.
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The Britannica Guide to Analysis and Calculus.pdf
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