Integral Rule of Secant function

Formula

$\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\log_{e}{|\sec{x}+\tan{x}|}+c$

The integral of secant is equal to the natural logarithm of sum of secant and tan functions

Introduction

Let $x$ be a variable and also represents an angle of a right triangle (or right-angled triangle). The secant of angle $x$ is written in mathematical form as $\sec{x}$ in trigonometry mathematics.

The indefinite integral of the secant of angle $x$ with respect to $x$ is written in mathematical form as follows.

$\displaystyle \int{\sec{x}}\,dx$

The indefinite integral of secant of angle $x$ is equal to the natural logarithm of secant of angle $x$ plus tan of angle $x$, and plus the constant of integration.

$\implies$ $\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\log_{e}{|\sec{x}+\tan{x}|}+c$

According to the logarithms, the natural logarithm is also written in the following form simply in mathematics.

$\implies$ $\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\ln{|\sec{x}+\tan{x}|}+c$

Alternative forms

The integral rule of secant function can be written in terms of any variable.

$(1).\,\,\,$ $\displaystyle \int{\sec{u}}\,du$ $\,=\,$ $\log_{e}{|\sec{u}+\tan{u}|}+c$

$(2).\,\,\,$ $\displaystyle \int{\sec{t}}\,dt$ $\,=\,$ $\log_{e}{|\sec{t}+\tan{t}|}+c$

$(3).\,\,\,$ $\displaystyle \int{\sec{y}}\,dy$ $\,=\,$ $\log_{e}{|\sec{y}+\tan{y}|}+c$

Proof

Learn how to prove the integral rule of secant function in integral calculus.