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The integral of secant cubed is a frequent and challenging [1] indefinite integral of elementary calculus : 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 ...
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 0 {\displaystyle 0} is the inverse Gudermannian function , gd − 1 . {\textstyle \operatorname {gd} ^{-1}.}
where sgn(x) is the sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive. This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
Other integrals. where. (Note that the value of the expression is independent of the value of n, which is why it does not appear in the integral.) where. and Γ (x,y) is the upper incomplete gamma function. when , , and. when , , and.
v. t. e. In mathematics (particularly multivariable calculus ), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding ...
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is. Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. [1]
Integrand involving both cosine and tangent. Integrand involving both sine and cotangent. Integrand involving both cosine and cotangent. Integrand involving both secant and tangent. Integrand involving both cosecant and cotangent. Integrals in a quarter period. Integrals with symmetric limits. Integral over a full circle.
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...