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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