Primitive of Arcsecant of x over a over x squared

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Theorem

$\ds \int \dfrac 1 {x^2} \arcsec \frac x a \rd x = \begin {cases}

-\dfrac 1 x \arcsec \dfrac x a + \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\ -\dfrac 1 x \arcsec \dfrac x a - \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end {cases}$


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \arcsec \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\

\dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end {cases}\)

Derivative of $\arcsec \dfrac x a$


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {x^2}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {-1} x\) Primitive of Power


First let $\arcsec \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.

Then:

\(\ds \int \dfrac 1 {x^2} \arcsec \frac x a\) \(=\) \(\ds \frac {-1} x \arcsec \frac x a - \int \frac {-1} x \paren {\frac a {x \sqrt {x^2 - a^2} } } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds -\dfrac 1 x \arcsec \dfrac x a + a \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds -\dfrac 1 x \arcsec \dfrac x a + a \paren {\frac {\sqrt {x^2 - a^2} } {a^2 x} } + C\) Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$
\(\ds \) \(=\) \(\ds -\dfrac 1 x \arcsec \dfrac x a + \frac {\sqrt {x^2 - a^2} } {a x} + C\) simplifying


Similarly, let $\arcsec \dfrac x a$ be in the interval $\openint {\dfrac \pi 2} \pi$.

Then:

\(\ds \int \dfrac 1 {x^2} \arcsec \frac x a\) \(=\) \(\ds \frac {-1} x \arcsec \frac x a - \int \frac {-1} x \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds -\dfrac 1 x \arcsec \dfrac x a - a \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds -\dfrac 1 x \arcsec \dfrac x a - a \paren {\frac {\sqrt {x^2 - a^2} } {a^2 x} } + C\) Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$
\(\ds \) \(=\) \(\ds -\dfrac 1 x \arcsec \dfrac x a - \frac {\sqrt {x^2 - a^2} } {a x} + C\) simplifying

$\blacksquare$


Also see


Sources

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