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.

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

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

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.

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

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.

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