Integral of the secant function

In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities, This formula is useful for evaluating various trigonometric integrals.

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

For numerical applications, all of the above expressions result in loss of significance for some arguments.

An alternative expression in terms of the inverse hyperbolic sine arsinh is numerically well behaved for real arguments

It is important because it is the vertical coordinate of the Mercator projection, used for marine navigation with constant compass bearing.

Three common expressions for the integral of the secant, are equivalent because Proof: we can separately apply the tangent half-angle substitution

to each of the three forms, and show them equivalent to the same expression in terms of

The conventional solution for the Mercator projection ordinate may be written without the absolute value signs since the latitude

, Let Therefore, The integral of the secant function was one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory.

[3] He applied his result to a problem concerning nautical tables.

[1] In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums.

[4] He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection.

[3] In the 1640s, Henry Bond, a teacher of navigation, surveying, and other mathematical topics, compared Wright's numerically computed table of values of the integral of the secant with a table of logarithms of the tangent function, and consequently conjectured that[3] This conjecture became widely known, and in 1665, Isaac Newton was aware of it.

This was the formula discovered by James Gregory.

[1] Although Gregory proved the conjecture in 1668 in his Exercitationes Geometricae,[7] the proof was presented in a form that renders it nearly impossible for modern readers to comprehend; Isaac Barrow, in his Lectiones Geometricae of 1670,[8] gave the first "intelligible" proof, though even that was "couched in the geometric idiom of the day.

"[3] Barrow's proof of the result was the earliest use of partial fractions in integration.

[3] Adapted to modern notation, Barrow's proof began as follows: Substituting u = sin θ, du = cos θ dθ, reduces the integral to Therefore, as expected.

The integral can also be derived by using a somewhat non-standard version of the tangent half-angle substitution, which is simpler in the case of this particular integral, published in 2013,[10] is as follows: Substituting: The integral can also be solved by manipulating the integrand and substituting twice.

Using the definition sec θ = ⁠1/cos θ⁠ and the identity cos2 θ + sin2 θ = 1, the integral can be rewritten as Substituting u = sin θ, du = cos θ dθ reduces the integral to The reduced integral can be evaluated by substituting u = tanh t, du = sech2 t dt, and then using the identity 1 − tanh2 t = sech2 t. The integral is now reduced to a simple integral, and back-substituting gives which is one of the hyperbolic forms of the integral.

A similar strategy can be used to integrate the cosecant, hyperbolic secant, and hyperbolic cosecant functions.

It is also possible to find the other two hyperbolic forms directly, by again multiplying and dividing by a convenient term: where

Substituting u = tan θ, du = sec2 θ dθ, reduces to a standard integral: where sgn is the sign function.

Likewise: Substituting u = |sec θ|, du = |sec θ| tan θ dθ, reduces to a standard integral: Under the substitution

Finally, if theta is real-valued, we can indicate this with absolute value brackets in order to get the equation into its most familiar form: The integral of the hyperbolic secant function defines the Gudermannian function: The integral of the secant function defines the Lambertian function, which is the inverse of the Gudermannian function: These functions are encountered in the theory of map projections: the Mercator projection of a point on the sphere with longitude λ and latitude ϕ may be written[11] as: D. T. Whiteside, editor, The Mathematical Papers of Isaac Newton, Cambridge University Press, 1967, volume 1, pages 466–467 and 473–475.