Use integration by parts to evaluate the integral $$ \int x e^{x} \, dx $$.

Question

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To evaluate the integral $$ \int x e^{x} \, dx $$ using integration by parts, we start by identifying the parts of the formula:

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

We choose:
- $$ u = x $$ (which means $$ du = dx $$)
- $$ dv = e^x \, dx $$ (which means $$ v = e^x $$)

Now we can apply the integration by parts formula:

1. Compute $$ uv $$:
   $$
   uv = x e^x
   $$

2. Compute $$ \int v \, du $$:
   $$
   \int v \, du = \int e^x \, dx = e^x
   $$

Now we can substitute these into the integration by parts formula:

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

Substituting the integral we computed:

$$
\int x e^x \, dx = x e^x - e^x + C
$$

where $$ C $$ is the constant of integration.

Finally, we can factor out $$ e^x $$:

$$
\int x e^x \, dx = e^x (x - 1) + C
$$

Thus, the evaluated integral is:

$$
\boxed{e^x (x - 1) + C}
$$
The integral $$ \int x e^{x} \, dx $$ evaluates to $$ e^x (x - 1) + C $$.

Use integration by parts to evaluate the integral \( \int x e^{x} \, dx \).