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 function cubed, which, despite seemingly special, comes up rather frequently in applications.
Proof that the different antiderivatives are equivalent
The integral of the secant function was one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory.
He applied his result to a problem concerning nautical tables.
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
This conjecture became widely known, and in 1665, Isaac Newton was aware of it.
By a standard substitution (Gregory's approach)
adding them gives
The integral is evaluated as follows:
This was the formula discovered by James Gregory.
By partial fractions and a substitution (Barrow's approach)
Although Gregory proved the conjecture in 1668 in his Exercitationes Geometricae, the proof was presented in a form that renders it nearly impossible for modern readers to comprehend; Isaac Barrow, in his Geometrical Lectures of 1670, gave the first "intelligible" proof, though even that was "couched in the geometric idiom of the day."
Barrow's proof of the result was the earliest use of partial fractions in integration.
Adapted to modern notation, Barrow's proof began as follows:
By the Weierstrass substitution
by the double-angle formulas.
As for the integral of the secant function,
The integral can also be derived by using the a somewhat non-standard version of the Weierstrass substitution, which is simpler in the case of this particular integral, published in 2013, is as follows:
Gudermannian and lambertian
The integral of the secant function defines the Lambertian function, which is the inverse of the Gudermannian function:
This is encountered in the theory of map projections: the Mercator projection of a point with longitude θ and latitude φ may be written as:
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Integral of the secant function.