Question

Evaluate the following:$\int (1+x)e^{x}dx$

Answer 1

$=\int (x+x{e}^x)dx$

$=\int xdx+\int x{e}^{x}dx$

$=\frac{x^2}{2}+\int x{e}^{x}dx$

Consider $\int x{e}^{x}dx$ by integrating by parts

Let $u=x\Rightarrow du=dx$ and $dv=e^{x}dx\Rightarrow v=e^x$

$=\frac{x^2}{2}+x{e}^x-\int e^{x}dx$

$=\frac{x^2}{2}+x{e}^x-e^x+c$ where $c$ is the constant of integration.