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Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx,} where R {\displaystyle R} is a rational function of x {\displaystyle x} and a x 2 + b x + c {\textstyle {\sqrt {ax^{2}+bx+c}}} .
The solutions in terms of the original variable are obtained by substituting x 3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested only in real solutions, the solutions of the original equation are
brings the Abel equation of the first kind to the canonical form u ′ = u 3 + ϕ ( ξ ) . {\displaystyle u'=u^{3}+\phi (\xi ).\,} Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation in an implicit form.
Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts , and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid solutions, and the other refers to interstitial solid solutions.
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.
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Approximate methods involve three basic steps: (1) counting the number of synonymous and nonsynonymous sites in the two sequences, or estimating this number by multiplying the sequence length by the proportion of each class of substitution; (2) counting the number of synonymous and nonsynonymous substitutions; and (3) correcting for multiple ...