Integral of sec²x formula
Formula
$\displaystyle \int{\sec^2{x}}\,dx \,=\, \tan{x}+c$
What is the integral of sec2x?
The integral of secant squared $x$ with respect to $x$ equals tangent of $x$ plus constant of integration.
When an angle is represented by the variable \(x\), the tangent function is written as $\tan{x}$, and the secant squared function is written as $\sec^2{x}$ in trigonometry; in calculus, these two trigonometric functions are directly related through differentiation and integration.
In integral calculus, the indefinite integral of $\sec^2{x}$ with respect to $x$ is written as follows.
$\displaystyle \int{\sec^2{x}}\,dx$
In differential calculus, the derivative of $\tan{x}$ with respect to $x$ is $\sec^2{x}$. Consequently, in integral calculus, the antiderivative of $\sec^2{x}$ with respect to $x$ is $\tan{x}+c$, where $c$ is the constant of integration.
$\displaystyle \int{\sec^2{x}}\,dx \,=\, \tan{x}+c$
In integral calculus, the secant function often appears in squared form when evaluating integrals of trigonometric functions. Therefore, the integral of $\sec^2{x}$ is a standard and widely used formula in calculus.
Examples of integral of sec2x?
Here are some worked examples to demonstrate how to apply the integral of $\sec^2{x}$ formula in calculus.
Example
$\displaystyle \int{\sec^2{(6x)}}\,dx$ $\,=\,$ $\dfrac{1}{6}\tan{x}+c$
Example
$\displaystyle \int{3\sec^2{(x)}}\,dx$ $\,=\,$ $3\tan{x}+c$
Example
$\displaystyle \int_{0}^{\frac{\pi}{4}}{\sec^2{(x)}}\,dx$ $\,=\,$ $1$
Example
$\displaystyle \int{\big(4+5\sec^2{(x)}\big)}\,dx$ $\,=\,$ $4x+5\tan{x}+c$
Alternative forms
In calculus, the integration of the secant squared function formula is also commonly written in the following form.
Example
$\displaystyle \int{\sec^2{\theta}}\,d\theta$ $\,=\,$ $\tan{\theta}+c$
You have learned the formula for $\int{\sec^2{x}}\,dx$ and its use in calculus. Next, let’s study the integral of the secant squared function with worked examples, proof and applications.
Proof of the Integral of Sec²x
Learn how to prove that the integral of secant squared $x$ equals tangent $x$ plus the constant of integration.