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Concept
Integral of the secant function
Summary
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.
The definite integral of the secant function starting from is the inverse Gudermannian function, 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
The integral of the secant function was historically one of the first integrals of its type ever evaluated, before most of the development of integral calculus. 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 Under this substitution and
First,
Second,
Third, using the tangent addition identity
So all three expressions describe the same quantity.
The conventional solution for the Mercator projection ordinate may be written without the absolute value signs since the latitude lies between and ,
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. He applied his result to a problem concerning nautical tables. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums. He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection.
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