Primitive of Power of x less one over Power of x minus Power of a
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Theorem
- $\ds \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$
Proof
\(\ds u\) | \(=\) | \(\ds x^n - a^n\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds n x^{n - 1}\) | Power Rule for Derivatives and Derivative of Constant | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {x^{n - 1} \rd x} {x^n - a^n}\) | \(=\) | \(\ds \frac 1 n \ln \size {x^n - a^n} + C\) | Primitive of Function under its Derivative |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^n \pm a^n$: $14.331$