Primitive of Function of a x + b

Theorem

$\displaystyle \int F \left({a x + b}\right) \ \mathrm d x = \frac 1 a \int F \left({u}\right) \ \mathrm d u$

where $u = a x + b$.

Proof

 $\displaystyle u$ $=$ $\displaystyle a x + b$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d u} {\mathrm d x}$ $=$ $\displaystyle a$ Derivative of Function of Constant Multiple: Corollary $\displaystyle \implies \ \$ $\displaystyle \int F \left({a x + b}\right) \ \mathrm d x$ $=$ $\displaystyle \int \frac {F \left({u}\right)} a \ \mathrm d u$ Primitive of Composite Function $\displaystyle$ $=$ $\displaystyle \frac 1 a \int F \left({u}\right) \ \mathrm d u$ Primitive of Constant Multiple of Function

$\blacksquare$