Primitive of Function of Exponential Function

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Theorem

$\ds \int \map F {e^{a x} } \rd x = \frac 1 a \int \frac {\map F u} u \rd u$

where $u = e^{a x}$.


Proof

\(\ds u\) \(=\) \(\ds e^{a x}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds a e^{a x}\) Derivative of Exponential of a x
\(\ds \) \(=\) \(\ds a u\) Definition of $u$
\(\ds \leadsto \ \ \) \(\ds \int \map F {e^{a x} } \rd x\) \(=\) \(\ds \int \frac {\map F u} {a u} \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\map F u} u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Sources