Primitive of Reciprocal of Power of x by Power of Power of x minus Power of a

Theorem

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

Proof

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

\(=\)

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

multiplying top and bottom by $a^n$

\(\ds \)

\(=\)

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

adding and subtracting $x^n$

\(\ds \)

\(=\)

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

Linear Combination of Primitives

\(\ds \)

\(=\)

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

simplification

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^n \pm a^n$: $14.333$