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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.55$