Find the integral of x e^x dx.

To find the integral of ( x e^x , dx ), we can use the technique of integration by parts. This method is particularly useful when dealing with products of functions, like ( x ) and ( e^x ).

We start by letting:

• ( u = x ) (which we will differentiate)

• ( dv = e^x , dx ) (which we will integrate)

Next, we need to compute ( du ) and ( v ):

• Differentiating ( u ) gives us ( du = dx ).

• Integrating ( dv ) yields ( v = e^x ).

We then apply the integration by parts formula, which states:

[

\int u , dv = uv - \int v , du

]

Substituting our choices into this formula, we have:

[

\int x e^x , dx = x e^x - \int e^x , dx

]

Now, we need to evaluate the integral on the right:

[

\int e^x , dx = e^x

]

Substituting this back into our earlier equation, we have:

[

\