Evaluate the following integrals:

Let $I=\int e^{x}\left(\log x+\frac{1}{x}\right) d x$

We know that

$\int e^{x}\left\{f(x)+f^{\prime}(x)\right\}=e^{x} f(x)+c$

Here,

$f(x)=\log x ; f^{\prime}(x)=\frac{1}{x}$

$\int e^{x}\left(\log x+\frac{1}{x}\right) d x=e^{x} \log x+c$