Primitive of Function of Root of a x + b

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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

$\blacksquare$


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