Primitive of Function of Nth Root of a x + b
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Theorem
- $\ds \int \map F {\sqrt [n] {a x + b} } \rd x = \frac n a \int u^{n - 1} \map F u \rd u$
where $u = \sqrt [n] {a x + b}$.
Proof
\(\ds u\) | \(=\) | \(\ds \sqrt [n] {a x + b}\) | ||||||||||||
\(\ds u\) | \(=\) | \(\ds \paren {a x + b}^{1/n}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac 1 {n \paren {\sqrt [n] {a x + b} }^{n - 1} } \map {\frac \d {\d x} } {a x + b}\) | Chain Rule for Derivatives, Derivative of Nth Root | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {n u^{n - 1} } \map {\frac \d {\d x} } {a x + b}\) | substituting for $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {n u^{n - 1} }\) | Derivative of Function of Constant Multiple: Corollary | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \map F {\sqrt [n] {a x + b} } \rd x\) | \(=\) | \(\ds \int \frac {n u^{n - 1} } a \map F u \rd u\) | Primitive of Composite Function | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac n a \int u^{n - 1} \map F u \rd u\) | Primitive of Constant Multiple of Function |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.51$