Primitive of Function of a x + b

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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

$\blacksquare$


Sources