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where is the inverse Gudermannian function, the integral of the secant function. There are a number of reasons why this particular antiderivative is worthy of special attention: The technique used for reducing integrals of higher odd powers of secant to lower ones is fully present in this, the simplest case. The other cases are done in the same ...
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.
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
In particular, it can be used to evaluate the integral of the secant cubed, which, though seemingly special, comes up rather frequently in applications. [1] The definite integral of the secant function starting from is the inverse Gudermannian function, .
In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point.
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin ′ ( a ) = cos( a ), meaning that the rate of change of sin( x ) at a particular angle x = a is given ...
In particular, the function Mittag-Leffler has a particular case , which is the exponential function, the Mittag-Leffler distribution of order is therefore an exponential distribution. However, for α ∈ ( 0 , 1 ) {\displaystyle \alpha \in (0,1)} , the Mittag-Leffler distributions are heavy-tailed .
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 ...