Primitive of Function of a x + b

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$

where $u = a x + b$.


Proof

\(\ds u\) \(=\) \(\ds a x + b\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds a\) Derivative of Function of Constant Multiple: Corollary
\(\ds \leadsto \ \ \) \(\ds \int \map F {a x + b} \rd x\) \(=\) \(\ds \int \frac {\map F u} a \d u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \frac 1 a \int \map F u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_Function_of_a_x_%2B_b&oldid=740780"