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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."
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions.Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated.
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:
In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived, a {\displaystyle a} and b {\displaystyle b} are solved for f ( u ) {\displaystyle f(u)} .
The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral +, the substitution + = + can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
To detect authenticity, a test strip is placed into a DNA sample solution and examined for the presence of a gene specific to Atlantic white shrimp, which are particularly prone to substitution.
Oklahoma began its substitution with 19 seconds left on the play clock and the Sooners were flagged for a delay-of-game penalty before they could snap the ball. Fourth-and-5 became fourth-and-10 ...
For tonight's meal, make one of these cozy, low-carb dinner recipes. Each dish is full of tasty seasonal produce like butternut squash, mushrooms and Brussels sprouts while having no more than 15 ...