Proof of Integral of sec²x formula

Take $x$ as a variable, and it also represents angle of a right triangle. According to trigonometry, the secant squared of angle $x$ is written as $\sec^2{x}$ in mathematical form. The indefinite integration of secant squared function with respect to $x$ is written mathematically as follows.

$\displaystyle \int{\sec^2{x} \,}dx$

Let’s start deriving the integral formula for secant squared of angle function in integration.

Derivative of Tan function

Write the differentiation of tan function formula for expressing the derivative of tangent function with respect to $x$ in mathematical form.

$\dfrac{d}{dx}{\, \tan{x}} \,=\, \sec^2{x}$

Inclusion of an Arbitrary constant

According to differential calculus, the derivative of a constant is zero. Therefore, it does not influence the differentiation of tan function even an arbitrary constant is added to tan function.

$\implies$ $\dfrac{d}{dx}{(\tan{x}+c)} \,=\, \sec^2{x}$

Integral of sec²x function

As per integral calculus, the collection of all primitives of $\sec^2{x}$ function is called the integration of $\sec^2{x}$ function. It is written in mathematical form as follows.

$\displaystyle \int{\sec^2{x} \,}dx$

In this case, the primitive or an antiderivative of $\sec^2{x}$ function is $\tan{x}$ and the constant of integration ($c$).

$\dfrac{d}{dx}{(\tan{x}+c)} = \sec^2{x}$ $\,\Longleftrightarrow\,$ $\displaystyle \int{\sec^2{x} \,}dx = \tan{x}+c$

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{\sec^2{x} \,}dx = \tan{x}+c$

Therefore, it is proved that the integration of secant squared of an angle function is equal to the sum of the tan function and constant of integration.