Primitive of Function of Root of a x + b

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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

$\blacksquare$


Sources