Primitive of Arccosecant of x over a
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Theorem
- $\ds \int \arccsc \frac x a \rd x = \begin {cases} x \arccsc \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ x \arccsc \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end {cases}$
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:
| \(\ds u\) | \(=\) | \(\ds \arccsc \frac x a\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}\) | Derivative of $\arccsc \dfrac x a$ |
and let:
| \(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds x\) | Primitive of Constant |
First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.
Then:
| \(\ds \int \arccsc \frac x a \rd x\) | \(=\) | \(\ds x \arccsc \frac x a - \int x \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccsc \frac x a + a \int \frac {\d x} {\sqrt {x^2 - a^2} } + C\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccsc \frac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C\) | Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ |
Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$.
Then:
| \(\ds \int \arccsc \frac x a \rd x\) | \(=\) | \(\ds x \arccsc \frac x a - \int x \paren {\frac a {x \sqrt {x^2 - a^2} } } \rd x + C\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccsc \frac x a - a \int \frac {\d x} {\sqrt {x^2 - a^2} } + C\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccsc \frac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C\) | Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.498$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions