Primitive of Power of x over Odd Power of x plus Odd Power of a

Theorem

\(\ds \int \frac {x^{p - 1} \rd x} {x^{2 m + 1} + a^{2 m + 1} }\)

\(=\)

\(\ds \frac {2 \paren {-1}^{p - 1} } {\paren {2 m + 1} a^{2 m - p + 1} } \sum_{k \mathop = 1}^m \sin \frac {2 k p \pi} {2 m + 1} \map \arctan {\frac {x + a \map \cos {\dfrac {2 k \pi} {2 m + 1} } } {a \map \sin {\dfrac {2 k \pi} {2 m + 1} } } }\)

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \frac {\paren {-1}^{p - 1} } {\paren {2 m + 1} a^{2 m - p + 1} } \sum_{k \mathop = 1}^m \cos \frac {2 k p \pi} {2 m + 1} \map \ln {x^2 + 2 a x \map \cos {\dfrac {2 k \pi} {2 m + 1} } + a^2}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \frac {\paren {-1}^{p - 1} \map \ln {x + a} } {\paren {2 m + 1} a^{2 m - p + 1} }\)

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 $x^n \pm a^n$: $14.337$