# Primitive of Function of Root of a x + b

## Theorem

$\ds \int \map F {\sqrt {a x + b} } \rd x = \frac 2 a \int u \map F u \rd u$

where $u = \sqrt {a x + b}$.

## Proof

 $\ds u$ $=$ $\ds \sqrt {a x + b}$ $\ds u$ $=$ $\ds \paren {a x + b}^{1/2}$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds \frac 1 2 \paren {a x + b}^{-1/2} \map {\frac \d {\d x} } {a x + b}$ Chain Rule for Derivatives, Power Rule for Derivatives $\ds$ $=$ $\ds \frac 1 {2 u} \map {\frac \d {\d x} } {a x + b}$ substituting for $u$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds \frac a {2 u}$ Derivative of Function of Constant Multiple: Corollary $\ds \leadsto \ \$ $\ds \int \map F {\sqrt {a x + b} } \rd x$ $=$ $\ds \int \frac {2 u} a \map F u \rd u$ Primitive of Composite Function $\ds$ $=$ $\ds \frac 2 a \int u \map F u \rd u$ Primitive of Constant Multiple of Function

$\blacksquare$