\[\int x{e}^{x} \, dx\]
1
Use Integration by Parts on ∫xex dx\int x{e}^{x} \, dx.
Let u=xu=x, dv=exdv={e}^{x}, du=dxdu=dx, v=exv={e}^{x}

2
Substitute the above into uv−∫v duuv-\int v \, du.
xex−∫ex dxx{e}^{x}-\int {e}^{x} \, dx

3
The integral of ex{e}^{x} is ex{e}^{x}.
xex−exx{e}^{x}-{e}^{x}

4
Add constant.
xex−ex+Cx{e}^{x}-{e}^{x}+C

Done
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