# 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, Integral of the secant function_sentence_0

This formula is useful for evaluating various trigonometric integrals. Integral of the secant function_sentence_1

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. Integral of the secant function_sentence_2

## Proof that the different antiderivatives are equivalent Integral of the secant function_section_0

### Hyperbolic forms Integral of the secant function_section_2

Let Integral of the secant function_sentence_3

Therefore, Integral of the secant function_sentence_4

## History Integral of the secant function_section_3

The integral of the secant function was one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. Integral of the secant function_sentence_5

He applied his result to a problem concerning nautical tables. Integral of the secant function_sentence_6

In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums. Integral of the secant function_sentence_7

He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection. Integral of the secant function_sentence_8

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 Integral of the secant function_sentence_9

This conjecture became widely known, and in 1665, Isaac Newton was aware of it. Integral of the secant function_sentence_10

## Evaluations Integral of the secant function_section_4

### By a standard substitution (Gregory's approach) Integral of the secant function_section_5

Starting with Integral of the secant function_sentence_11

adding them gives Integral of the secant function_sentence_12

The integral is evaluated as follows: Integral of the secant function_sentence_13

as claimed. Integral of the secant function_sentence_14

This was the formula discovered by James Gregory. Integral of the secant function_sentence_15

### By partial fractions and a substitution (Barrow's approach) Integral of the secant function_section_6

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." Integral of the secant function_sentence_16

Barrow's proof of the result was the earliest use of partial fractions in integration. Integral of the secant function_sentence_17

Adapted to modern notation, Barrow's proof began as follows: Integral of the secant function_sentence_18

Therefore, Integral of the secant function_sentence_19

as expected. Integral of the secant function_sentence_20

### By the Weierstrass substitution Integral of the secant function_section_7

#### Standard Integral of the secant function_section_8

Hence, Integral of the secant function_sentence_21

by the double-angle formulas. Integral of the secant function_sentence_22

As for the integral of the secant function, Integral of the secant function_sentence_23

as before. Integral of the secant function_sentence_24

#### Non-standard Integral of the secant function_section_9

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: Integral of the secant function_sentence_25

## Gudermannian and lambertian Integral of the secant function_section_10

The integral of the secant function defines the Lambertian function, which is the inverse of the Gudermannian function: Integral of the secant function_sentence_26

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: Integral of the secant function_sentence_27