What did I do wrong when integrating by parts?

Question

I am attempting to solve the following:

$$\int e^xx \ dx$$

This looks like it should be solved using integration by parts. Using an online calculator, it chose $f$ to be $x$ and $g$ to be $e^x$ so that what would remain is the following:

$$\int e^xx \ dx = x\ e^x - \int e^x \ dx$$

However, I, instead, chose $g'$ to be $x$, resultatively $g$ to be $1$ and $f$ to be $e^x$. When computing my result I ended up with:

$$\int e^xx \ dx = e^x - \int e^x \ dx$$

Other than the fact that this isn't the proper way to solve this integral since... it's wrong, what did I do wrong here? Rather, what about my thought process was wrong? I should be able to pick $e^x$ or $x$ as $f$ or $g$ as long a I'm consistent due to associativity, so what's the error?

Answer 1

Note:

$g'=x$
$g=\frac{x^2}{2}+C$

This is where you went wrong