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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.
This is in general good practice, but may require that the original author of Ellipse anticipated such a problem, and defined the mutators as returning a value. Also, it requires the client code to test the return value for support of the stretch function, which in effect is like testing if the referenced object is either a circle or an ellipse.
Alternatives assessment or alternatives analysis is a problem-solving approach used in environmental design, technology, and policy.It aims to minimize environmental harm by comparing multiple potential solutions in the context of a specific problem, design goal, or policy objective.
Liskov substitution principle imposes some standard requirements on signatures that have been adopted in newer object-oriented programming languages (usually at the level of classes rather than types; see nominal vs. structural subtyping for the distinction): Contravariance of method parameter types in the subtype.
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 problem is that the free variable x of t became bound during the substitution. The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z , so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.
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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."
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