Primitive of Function of Nth Root of a x + b

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 $u = \sqrt [n] {a x + b}$.

$\ds u$

$=$

$\ds \sqrt [n] {a x + b}$

$\ds u$

$=$

$\ds \paren {a x + b}^{1/n}$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds \frac 1 {n \paren {\sqrt [n] {a x + b} }^{n - 1} } \map {\frac \d {\d x} } {a x + b}$

$\ds $

$=$

$\ds \frac 1 {n u^{n - 1} } \map {\frac \d {\d x} } {a x + b}$

substituting for $u$

$\ds $

$=$

$\ds \frac a {n u^{n - 1} }$

$\ds \leadsto \ \ $

$\ds \int \map F {\sqrt [n] {a x + b} } \rd x$

$=$

$\ds \int \frac {n u^{n - 1} } a \map F u \rd u$

$\ds $

$=$

$\ds \frac n a \int u^{n - 1} \map F u \rd u$

$\blacksquare$