Primitive of Power of x over Even Power of x minus Even Power of a
Jump to navigation
Jump to search
Theorem
\(\ds \int \frac {x^{p - 1} \rd x} {x^{2 m} - a^{2 m} }\) | \(=\) | \(\ds \frac 1 {2 m a^{2 m - p} } \sum_{k \mathop = 1}^{m - 1} \map \cos {\frac {k p \pi} m} \map \ln {x^2 - 2 a x \map \cos {\frac {k \pi} m} + a^2}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac 1 {m a^{2 m - p} } \sum_{k \mathop = 1}^{m - 1} \map \sin {\frac {k p \pi} m} \map \arctan {\frac {x - a \map \cos {\dfrac {k \pi} m} } {a \map \sin {\dfrac {k \pi} m} } }\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {2 m a^{2 m - p} } \paren {\map \ln {x - a} + \paren {-1}^p \map \ln {x + a} }\) |
where $0 < p \le 2 m$.
Proof
The integrand is a rational function.
It has simple poles at $x = \omega_k a$ where $\omega_k = e^{\pi i k /m} $, $k = 0, 1, \ldots, 2 m - 1$ are the $2m$'th roots of unity:
\(\ds \map f x\) | \(=\) | \(\ds \dfrac {x^{p - 1} } {x^{2 m} - a^{2 m} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x^{p - 1} } {\ds \prod_{k \mathop = 0}^{2 m - 1} \paren {x - \omega_k a} }\) |
Its residue at $x = \omega_k$ is then:
- $\Res f {\omega_k a} = \dfrac {\paren {\omega_k a}^{p - 2m} } {2 m}$
so that:
- $\ds \map f x = \sum_{k \mathop = 0}^{2 m - 1} \dfrac {\paren {\omega_k a}^{p - 2m} } {2 m \paren {x - \omega_k a} }$
which has a primitive:
- $\ds \map F x = \sum_{k \mathop = 0}^{2 m - 1} \dfrac {\paren {\omega_k a}^{p - 2m} } {2 m} \map \ln {x - \omega_k a}$
For positive real $x, a \in \R_{>0}$ we can write:
- $\map \ln {x - \omega_k a} = \dfrac 1 2 \map \ln {x^2 - 2 a x \map \cos {\dfrac {2 \pi k} m} + a^2} - i \map \arctan {\dfrac {a \map \sin {\dfrac {2 \pi k} m} } {x - a \map \cos {\dfrac {2 \pi k} m} } }$
- $\paren {\omega_k a}^{p - 2 m} = a^{p - 2 m} \paren {\map \cos {\dfrac {p k \pi} m} + i \map \sin {\dfrac {p k \pi} m} }$
and the formula follows.
This needs considerable tedious hard slog to complete it. In particular: Fill in the details To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^n \pm a^n$: $14.336$
- Robert Israel (https://math.stackexchange.com/users/8508/robert-israel), Indefinite Integral of $\dfrac {x^{p-1} } {x^{2m} - a^{2m} }$, URL (version: 2020-12-30): https://math.stackexchange.com/q/3966353