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The inverse tangent integral is defined by: = The arctangent is taken to be the principal branch; that is, − π /2 < arctan(t) < π /2 for all real t. [1]Its power series representation is
There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin −1, asin, or, as is used on this page, arcsin. For each inverse trigonometric integration formula below there is a corresponding formula in the list of integrals of inverse hyperbolic functions.
The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals.
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. [1] (This convention is used throughout this article.)
Lists of integrals; Integral transform; Leibniz integral rule; Definitions; Antiderivative; Integral Riemann integral; Lebesgue integration; Contour integration; Integral of inverse functions; Integration by; Parts; Discs; Cylindrical shells; Substitution (trigonometric, tangent half-angle, Euler) Euler's formula; Partial fractions (Heaviside's ...
Since and the inverse function : are continuous, they have antiderivatives by the fundamental theorem of calculus. Laisant proved that if F {\displaystyle F} is an antiderivative of f {\displaystyle f} , then the antiderivatives of f − 1 {\displaystyle f^{-1}} are:
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 standard method of evaluating the secant integral presented in various references involves multiplying the numerator and denominator by sec θ + tan θ and then using the substitution u = sec θ + tan θ. This substitution can be obtained from the derivatives of secant and tangent added together, which have secant as a common factor. [6]