# Proof for Integration of Secant function

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

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

Let’s learn how to prove the integration of the secant function in integral calculus from understandable procedure.

### Prepare the secant function for integration

The integration of secant function cannot be calculated by the integration by parts. So, it requires a special process to find its integral.

$=\,\,\,$ $\displaystyle \int{(\sec{x} \times 1)}\,dx$

Multiply the secant function by the sum of secant and tan functions and then divide it by the same.

$=\,\,\,$ $\displaystyle \int{\bigg(\sec{x} \times \dfrac{\sec{x}+\tan{x}}{\sec{x}+\tan{x}}\bigg)}\,dx$

Now, multiply the secant function with the fractional function.

$=\,\,\,$ $\displaystyle \int{\dfrac{\sec{x} \times (\sec{x}+\tan{x})}{\sec{x}+\tan{x}}}\,dx$

$=\,\,\,$ $\displaystyle \int{\dfrac{\sec{x} \times \sec{x}+\sec{x} \times \tan{x}}{\sec{x}+\tan{x}}}\,dx$

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

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

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

### Transform Rational function by differentiation

Suppose $u \,=\, \sec{x}+\tan{x}$

Now, differentiate the equation with respect to $x$.

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

Use the sum rule of the derivatives to find the derivative of sum of the two trigonometric functions secant and tangent.

$\implies$ $\dfrac{du}{dx}$ $\,=\,$ $\dfrac{d}{dx}{\sec{x}}$ $+$ $\dfrac{d}{dx}{\tan{x}}$

According to the derivative rule of secant and differentiation rule of tan function, find the derivatives of the secant and tan functions with respect to $x$.

$\implies$ $\dfrac{du}{dx}$ $\,=\,$ $\sec{x}\tan{x}$ $+$ $\sec^2{x}$

$\,\,\,\therefore\,\,\,\,\,\,$ $du$ $\,=\,$ $(\sec{x}\tan{x}+\sec^2{x}) \times dx$

The expression in the denominator is considered to denote by a variable $u$ and the expression in the numerator can be replaced by the differential $du$.

$\implies$ $\displaystyle \int{\dfrac{(\sec{x}\tan{x}+\sec^2{x}) \times dx}{\sec{x}+\tan{x}}}$ $\,=\,$ $\displaystyle \int{\dfrac{du}{u}}$

Thus, the integral function in terms of $x$ is converted as an integral of function in terms of $u$.

### Find the Integration of Reciprocal function

It is time for the integration of the function.

$=\,\,\,$ $\displaystyle \int{\dfrac{1 \times du}{u}}$

$=\,\,\,$ $\displaystyle \int{\dfrac{1}{u}} \times du$

$=\,\,\,$ $\displaystyle \int{\dfrac{1}{u}}\,du$

The integral of the reciprocal of the variable can be calculated by reciprocal integration rule.

$=\,\,\,$ $\log_{e}{|u|}+c$

Now, replace the value of $u$ for finishing the process of finding the integral of the secant function.

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

The natural logarithmic function can also be written as follows as per the logarithms.

$=\,\,\,$ $\ln{|\sec{x}+\tan{x}|}+c$

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