Primitive of Power of x over Odd Power of x minus Odd Power of a
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Theorem
\(\ds \int \frac {x^{p - 1} \rd x} {x^{2 m + 1} - a^{2 m + 1} }\) | \(=\) | \(\ds \frac {-2} {\paren {2 m + 1} a^{2 m - p + 1} } \sum_{k \mathop = 1}^m \sin \frac {2 k p \pi} {2 m + 1} \map \arctan {\frac {x - a \map \cos {\dfrac {2 k \pi} {2 m + 1} } } {a \map \sin {\dfrac {2 k \pi} {2 m + 1} } } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {\paren {2 m + 1} a^{2 m - p + 1} } \sum_{k \mathop = 1}^m \cos \frac {2 k p \pi} {2 m + 1} \map \ln {x^2 - 2 a x \cos \frac {2 k \pi} {2 m + 1} + a^2}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {\map \ln {x - a} } {\paren {2 m + 1} a^{2 m - p + 1} }\) |
where $0 < p \le 2 m + 1$.
Proof
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Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^n \pm a^n$: $14.338$