Integral of Secant (∫sec x dx) with Examples
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Formula
$\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\ln{|\sec{x}+\tan{x}|}+c$
What is the Integral of Secant?
The integral of secant $x$ with respect to $x$ is equal to the natural logarithm of the absolute value of secant $x$ plus tangent $x$, plus a constant of integration.
The integral of secant is the antiderivative of the trigonometric function $\sec{x}$ with respect to $x$. It represents the function whose derivative is $\sec{x}$.
$\displaystyle \int{\sec{x}}\,dx$
Unlike many basic trigonometric integrals, the integral of secant cannot be evaluated directly and requires a special method of integration. The result is expressed in logarithmic form and involves both $\sec{x}$ and $\tan{x}$, along with a constant of integration.
$\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\log_{e}{|\sec{x}+\tan{x}|}+c$
This integral is commonly used in calculus and appears in applications involving trigonometric identities and advanced integration techniques.
Popular Forms of the Integral of Secant
The integral of secant can be written in two equivalent forms, all giving the same result but appearing differently in textbooks or online references. Knowing these variations helps solve calculus problems and recognize trigonometric identities.
Form: 1
$\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\ln{\bigg|\tan{\bigg(\dfrac{\pi}{4}+\dfrac{x}{2}\bigg)}\bigg|}$ $+$ $c$
It is a special form of the secant integral expressed in tangent half-angle and written in natural logarithm form.
Form: 2
$\displaystyle \int{\sec{x}}\,dx$ $\,=\,$ $\ln{\bigg|\dfrac{1+\sin{x}}{\cos{x}}\bigg|}$ $+$ $c$
It is a special form of the secant integral written in sine and cosine functions and written in natural logarithm form.
Proof of the Integral of Secant
A rigorous mathematical proof showing how the integral of secant, ∫sec x dx, is derived and expressed in natural logarithmic form.