Primitive of Function of Exponential Function

Theorem

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

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

$\ds u$

$=$

$\ds e^{a x}$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds a e^{a x}$

$\ds $

$=$

$\ds a u$

Definition of $u$

$\ds \leadsto \ \ $

$\ds \int \map F {e^{a x} } \rd x$

$=$

$\ds \int \frac {\map F u} {a u} \rd u$

$\ds $

$=$

$\ds \frac 1 a \int \frac {\map F u} u \rd u$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.55$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: Important Transformations: $16.55.$