Primitive of Power of x by Arccosecant of x over a/Proof 2
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Theorem
- $\ds \int x^m \arccsc \frac x a \rd x = \begin{cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a - \dfrac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
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:
| \(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x^m\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1}\) | Primitive of Power |
First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.
Then:
| \(\ds \int x^m \arccsc \frac x a \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \int \frac {x^{m + 1} } {m + 1} \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arccsc \frac x a + \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} }\) | Primitive of Constant Multiple of Function |
Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$.
Then:
| \(\ds \int x^m \arccsc \frac x a \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \int \frac {x^{m + 1} } {m + 1} \paren {\frac a {x \sqrt {x^2 - a^2} } } \rd x\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} }\) | Primitive of Constant Multiple of Function |
$\blacksquare$