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 $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {2 u}\) | 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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.50$