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The integral of secant cubed is a frequent and challenging [1] indefinite integral of elementary calculus:
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}.}
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
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.
For a definite integral, the bounds change once the substitution is performed and are determined using the equation = , with values in the range < <. Alternatively, apply the boundary terms directly to the formula for the antiderivative.
Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.
The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.