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In calculus, integration by parametric derivatives, also called parametric integration, [1] is a method which uses known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution.
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the dependent variables are x and y and are given by parametric equations in t).
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and published in his Principia in 1687, [2] which was the first problem in the field to be clearly formulated and correctly solved, and was one of the most difficult problems tackled by variational methods prior to the twentieth century.
The above definition of an integral curve α for a vector field X, passing through p at time t 0, is the same as saying that α is a local solution to the ordinary differential equation/initial value problem =; ′ = (()).
The method also is applicable to other multiple integrals. [1] [2] Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, a double integral can be reduced to a single integration, as illustrated next. Reduction to a single integration makes a numerical evaluation much easier and more efficient.
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."