Answer
$\int\frac{e^{x}}{x}dx=C+ln|x|+\Sigma_{n=1}^\infty\frac{x^{n}}{n(n!)}$
Work Step by Step
$\int\frac{e^{x}}{x}dx=\int \frac{1}{x}\Sigma_{n=0}^\infty\frac{x^{n}}{n!}dx=\int \Sigma_{n=0}^\infty\frac{x^{n-1}}{n!}dx$
$\int \Sigma_{n=0}^\infty\frac{x^{n-1}}{n!}dx=\int \frac{1}{x}+\int \Sigma_{n=1}^\infty\frac{x^{n-1}}{n!}dx$
$=C+ln|x|+\Sigma_{n=1}^\infty \int \frac{x^{n-1}}{n!}dx$
$=C+ln|x|+\Sigma_{n=1}^\infty\frac{x^{n}}{n(n!)}$
Hence, $\int\frac{e^{x}}{x}dx=C+ln|x|+\Sigma_{n=1}^\infty\frac{x^{n}}{n(n!)}$
