Search results
Results from the WOW.Com Content Network
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials.It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be ...
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of () is the yellow (−) area subtracted from the blue (+) area
To solve an inexact differential equation, it may be transformed into an exact differential equation by finding an integrating factor. [2] Multiplying the original equation by the integrating factor gives: + =.
This visualization also explains why integration by parts may help find the integral of an inverse function f −1 (x) when the integral of the function f(x) is known. Indeed, the functions x(y) and y(x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx.
The difference between initial and final states of the system's internal energy does not account for the extent of the energy interactions transpired. Therefore, internal energy is a state function (i.e. exact differential), while heat and work are path functions (i.e. inexact differentials) because integration must account for the path taken.
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 ...