Integral of -x*exp(x) dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = - x$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$.

   Then $\operatorname{du}{\left(x \right)} = -1$.

   To find $v{\left(x \right)}$:

   1. The integral of the exponential function is itself.

      $$\int e^{x}\, dx = e^{x}$$

   Now evaluate the sub-integral.

2. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \left(- e^{x}\right)\, dx = - \int e^{x}\, dx$$

   1. The integral of the exponential function is itself.

      $$\int e^{x}\, dx = e^{x}$$

   So, the result is: $- e^{x}$

3. Now simplify:

   $$\left(1 - x\right) e^{x}$$

4. Add the constant of integration:

   $$\left(1 - x\right) e^{x}+ \mathrm{constant}$$

The answer is:

$$\left(1 - x\right) e^{x}+ \mathrm{constant}$$