Primitive of Exponential of a x over x
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Theorem
For $x > 0$:
\(\ds \int \frac {e^{a x} \rd x} x\) | \(=\) | \(\ds \ln x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln x + \dfrac {a x} {1 \times 1!} + \dfrac {\paren {a x}^2} {2 \times 2!} + \dfrac {\paren {a x}^3} {3 \times 3!} + \cdots + C\) |
Proof
\(\ds \int \frac {e^{a x} \rd x} x\) | \(=\) | \(\ds \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {a x}^k} {k!} } \rd x\) | Power Series Expansion for Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac 1 x \paren {1 + \sum_{k \mathop = 1}^\infty \frac {\paren {a x}^k} {k!} } \rd x\) | extracting the case $k = 0$ from the expansion | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \int \frac {\d x} x + \sum_{k \mathop = 1}^\infty \frac {a^k} {k!} \int x^{k - 1} \rd x\) | Linear Combination of Primitives | ||||||||||
\(\ds \) | \(=\) | \(\ds \ln x + \sum_{k \mathop = 1}^\infty \frac {a^k} {k!} \int x^{k - 1} \rd x\) | Primitive of Reciprocal: $x > 0$ so negative argument does not apply | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln x + \sum_{k \mathop = 1}^\infty \frac {a^k} {k!} \frac {x^k} k + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C\) | simplification |
The validity of $(1)$ follows from absolute convergence of the power series expansion.
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.513$