Primitive of Power of x over Power of Power of x minus Power of a

Theorem

$\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r} = a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }$

Proof

\(\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r}\)

\(=\)

\(\ds \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}\)

\(\ds \)

\(=\)

\(\ds \int \frac {x^{m - n} \paren {x^n - a^n + a^n} \rd x} {\paren {x^n - a^n}^r}\)

\(\ds \)

\(=\)

\(\ds \int \frac {x^{m - n} \paren {x^n - a^n} \rd x} {\paren {x^n - a^n}^r} + a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r}\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }\)

simplification

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^n \pm a^n$: $14.332$