# Primitive of Function of Root of a x + b

## Theorem

$\displaystyle \int F \left({\sqrt {a x + b}}\right) \ \mathrm d x = \frac 2 a \int u \ F \left({u}\right) \ \mathrm d u$

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

## Proof

 $\displaystyle u$ $=$ $\displaystyle \sqrt {a x + b}$ $\displaystyle u$ $=$ $\displaystyle \left({a x + b}\right)^{1/2}$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d u} {\mathrm d x}$ $=$ $\displaystyle \frac 1 2 \left({a x + b}\right)^{-1/2} \frac {\mathrm d} {\mathrm d x} \left({a x + b}\right)$ Chain Rule for Derivatives, Power Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \frac 1 {2 u} \frac {\mathrm d} {\mathrm d x} \left({a x + b}\right)$ substituting for $u$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d u} {\mathrm d x}$ $=$ $\displaystyle \frac a {2 u}$ Derivative of Function of Constant Multiple: Corollary $\displaystyle \implies \ \$ $\displaystyle \int F \left({\sqrt {a x + b} }\right) \ \mathrm d x$ $=$ $\displaystyle \int \frac {2 u} a F \left({u}\right) \ \mathrm d u$ Primitive of Composite Function $\displaystyle$ $=$ $\displaystyle \frac 1 a \int u \ F \left({u}\right) \ \mathrm d u$ Primitive of Constant Multiple of Function

$\blacksquare$