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In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial:
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution .
In cryptography, an S-box (substitution-box) is a basic component of symmetric key algorithms which performs substitution. In block ciphers , they are typically used to obscure the relationship between the key and the ciphertext , thus ensuring Shannon's property of confusion .
Often, theory can establish the existence of a change of variables, although the formula itself cannot be explicitly stated. For an integrable Hamiltonian system of dimension , with ˙ = / and ˙ = /, there exist integrals .
In cryptography, a substitution cipher is a method of encrypting in which units of plaintext are replaced with the ciphertext, in a defined manner, with the help of a key; the "units" may be single letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth. The receiver deciphers the text by performing ...
In optical fiber technology, the substitution method is a method of measuring the transmission loss of a fiber. It consists of: using a stable optical source, at the wavelength of interest, to drive a mode scrambler, the output of which overfills (drives) a 1 to 2 meter long reference fiber having physical and optical characteristics matching those of the fiber under test,
Euler substitution is a method for evaluating integrals of the form (, + +), where is a rational function of and + +. In such cases, the integrand can be changed to a rational function by using the substitutions of Euler.