Integral Resulting in Arcsecant

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Theorem

$\displaystyle \int \frac 1 {x\sqrt{x^2 - a^2} }\ \mathrm dx = \begin{cases} \dfrac 1 {|a|} \text {arcsec} \dfrac x {|a|} + C& : x > |a| \\ -\dfrac {1} {|a|} \text {arcsec} \dfrac x {|a|} + C& : x < -|a| \end{cases}$

where $a$ is a constant.

Proof

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \int \frac 1 {x\sqrt{x^2 - a^2} } \ \mathrm dx$	$=$	$\displaystyle \int \frac {1}{x\sqrt{a^2 \left({\frac {x^2}{a^2}-1 }\right)} } \ \mathrm dx$	$\displaystyle $	$\displaystyle $	$\displaystyle $	factor $a^2$ out of the radicand
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \int \frac 1 {x\sqrt{a^2} \sqrt{\left({\frac x a}\right)^2 - 1 } } \ \mathrm dx$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {\left \vert {a} \right \vert} \int \frac {1}{x\sqrt{\left({\frac x a}\right)^2 - 1} } \ \mathrm dx$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Substitute:

$\sec \theta = \dfrac x {|a|} \iff |a|\sec \theta = x$

for $\theta \in \left({0 .. \dfrac \pi 2}\right)\cup\left({\dfrac \pi 2 .. \pi}\right)$.

This substitution is valid for all $\dfrac {x}{|a|} \in \R \setminus \left({-1 .. 1}\right)$.

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By hypothesis:

$\left(x > |a|\right) \lor \left(x < -|a|\right)$

$\iff \left(\dfrac {x}{|a|} > 1\right) \lor \left(\dfrac {x}{|a|} < -1\right)$

... so this substitution will not change the domain of the integrand.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left \vert{a}\right \vert \sec \theta$	$=$	$\displaystyle x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from above
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \left \vert {a}\right \vert \sec \theta \tan \theta \frac {\mathrm d \theta}{\mathrm dx}$	$=$	$\displaystyle 1$	$\displaystyle $	$\displaystyle $	$\displaystyle $	differentiate WRT $x$, Derivative of Secant Function, Chain Rule
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \int \frac 1 {x\sqrt{x^2 - a^2} } \ \mathrm dx$	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\left \vert {a}\right \vert \sec \theta \tan \theta}{\left \vert {a}\right \vert \sec \theta \sqrt{\sec^2\theta - 1} } \frac {\mathrm d \theta}{\mathrm dx} \mathrm dx$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from above
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta}{\sqrt{\sec^2\theta - 1} }\ \mathrm d \theta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Integration by Substitution
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta}{\sqrt{\tan^2 \theta}\ \mathrm d \theta}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	corollary to sum of squares of sine and cosine
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta} {\left \vert \tan \theta \right \vert} \ \mathrm d \theta$	$\displaystyle $	$\displaystyle $	$\displaystyle $

By Shape of Tangent Function and the stipulated definition of $\theta$:

$(A) \quad \dfrac {x}{|a|} > 1 \iff \theta \in \left({0 .. \dfrac \pi 2}\right)$

and

$(B) \quad \dfrac {x}{|a|} < -1 \iff \theta \in \left({\dfrac \pi 2 .. \pi}\right)$

If $(A)$:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta} {\left \vert \tan \theta \right \vert} \ \mathrm d \theta$	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \mathrm d \theta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	definition of absolute value
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \theta + C$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Integration of a Constant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \text{arcsec} \frac {x} {\left \vert {a} \right \vert} + C$	$\displaystyle $	$\displaystyle $	$\displaystyle $	definition of arcsecant

If $(B)$:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta} {\left \vert \tan \theta \right \vert} \ \mathrm d \theta$	$=$	$\displaystyle \frac {1} {\left \vert {a} \right \vert} \int -1 \ \mathrm d \theta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	definition of absolute value
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle -\frac {1} {\left \vert {a} \right \vert} \theta + C$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Integration of a Constant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle -\frac {1} {\left \vert {a} \right \vert} \text{arcsec} \frac {x} {\left \vert {a} \right \vert} + C$	$\displaystyle $	$\displaystyle $	$\displaystyle $	definition of arcsecant

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Also see

Derivative of Arcsecant Function

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \int \frac 1 {x\sqrt{x^2 - a^2} } \ \mathrm dx\)	\(=\)	\(\displaystyle \int \frac {1}{x\sqrt{a^2 \left({\frac {x^2}{a^2}-1 }\right)} } \ \mathrm dx\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	factor $a^2$ out of the radicand
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \int \frac 1 {x\sqrt{a^2} \sqrt{\left({\frac x a}\right)^2 - 1 } } \ \mathrm dx\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {\left \vert {a} \right \vert} \int \frac {1}{x\sqrt{\left({\frac x a}\right)^2 - 1} } \ \mathrm dx\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left \vert{a}\right \vert \sec \theta\)	\(=\)	\(\displaystyle x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from above
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \left \vert {a}\right \vert \sec \theta \tan \theta \frac {\mathrm d \theta}{\mathrm dx}\)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	differentiate WRT $x$, Derivative of Secant Function, Chain Rule
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \int \frac 1 {x\sqrt{x^2 - a^2} } \ \mathrm dx\)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\left \vert {a}\right \vert \sec \theta \tan \theta}{\left \vert {a}\right \vert \sec \theta \sqrt{\sec^2\theta - 1} } \frac {\mathrm d \theta}{\mathrm dx} \mathrm dx\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from above
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta}{\sqrt{\sec^2\theta - 1} }\ \mathrm d \theta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integration by Substitution
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta}{\sqrt{\tan^2 \theta}\ \mathrm d \theta}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	corollary to sum of squares of sine and cosine
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta} {\left \vert \tan \theta \right \vert} \ \mathrm d \theta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta} {\left \vert \tan \theta \right \vert} \ \mathrm d \theta\)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \mathrm d \theta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of absolute value
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \theta + C\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integration of a Constant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \text{arcsec} \frac {x} {\left \vert {a} \right \vert} + C\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of arcsecant

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int \frac {\tan \theta} {\left \vert \tan \theta \right \vert} \ \mathrm d \theta\)	\(=\)	\(\displaystyle \frac {1} {\left \vert {a} \right \vert} \int -1 \ \mathrm d \theta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of absolute value
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -\frac {1} {\left \vert {a} \right \vert} \theta + C\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integration of a Constant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -\frac {1} {\left \vert {a} \right \vert} \text{arcsec} \frac {x} {\left \vert {a} \right \vert} + C\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of arcsecant

Integral Resulting in Arcsecant

Theorem

Proof

Also see

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