Primitive of Reciprocal of Power of x by Power of Power of x plus Power of a
Jump to navigation
Jump to search
Theorem
- $\ds \int \frac {\d x} {x^m \paren {x^n + a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n + a^n}^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n + a^n}^r}$
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 $a^n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x^m \paren {x^n + a^n}^r}\) | adding and subtracting $x^n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^n} \int \frac {\paren {x^n + a^n} \rd x} {x^m \paren {x^n + a^n}^r} - \frac 1 {a^n} \int \frac {x^n \rd x} {x^m \paren {x^n + a^n}^r}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n + a^n}^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n + a^n}^r}\) | simplification |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^n \pm a^n$: $14.328$