Primitive of Exponential of a x over x

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Theorem

$\displaystyle \int \frac {e^{a x} \rd x} x = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C$


Proof

\(\displaystyle \int \frac {e^{a x} \rd x} x\) \(=\) \(\displaystyle \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {a x}^k} {k!} } \rd x\) Power Series Expansion for Exponential Function
\(\displaystyle \) \(=\) \(\displaystyle \int \frac 1 x \paren {1 + \sum_{k \mathop = 1}^\infty \frac {\paren {a x}^k} {k!} } \rd x\) extracting the case $k = 0$ from the expansion
\((1):\quad\) \(\displaystyle \) \(=\) \(\displaystyle \int \frac {\d x} x + \sum_{k \mathop = 1}^\infty \frac {a^k} {k!} \int x^{k - 1} \rd x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \ln \size x + \sum_{k \mathop = 1}^\infty \frac {a^k} {k!} \int x^{k - 1} \rd x\) Primitive of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \ln \size x + \sum_{k \mathop = 1}^\infty \frac {a^k} {k!} \frac {x^k} k + C\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C\) simplification

The validity of $(1)$ follows from absolute convergence of the power series expansion.

$\blacksquare$


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