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resolver  aplicando Resolver los siguientes ejercicios de integrales impropias. 

The given integral is an improper integral:

\[
\int_0^{\infty} x e^{-x} \, dx
\]

To solve this, we can apply integration by parts. Let's set it up step by step.

### Step 1: Choose functions for integration by parts.

We know the formula for integration by parts is:

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

Choose:
- \( u = x \) so that \( du = dx \)
- \( dv = e^{-x} \, dx \) so that \( v = -e^{-x} \)

### Step 2: Apply the integration by parts formula.

\[
\int_0^{\infty} x e^{-x} \, dx = \left[ -x e^{-x} \right]_0^{\infty} + \int_0^{\infty} e^{-x} \, dx
\]

### Step 3: Evaluate the boundary terms.

For the first term:
\[
\lim_{x \to \infty} (-x e^{-x}) = 0 \quad \text{and} \quad (-x e^{-x}) \Big|_{x=0} = 0
\]
So, the boundary terms cancel out.

Now, we are left with the second integral:
\[
\int_0^{\infty} e^{-x} \, dx
\]

### Step 4: Solve the remaining integral.

This is a basic exponential integral:
\[
\int_0^{\infty} e^{-x} \, dx = \left[ -e^{-x} \right]_0^{\infty} = 1
\]

### Final Answer:
\[
\int_0^{\infty} x e^{-x} \, dx = 1
\]

Would you like any further details on this solution?

Here are some related questions to consider:

1. How would you solve \(\int_0^{\infty} x^2 e^{-x} \, dx\)?
2. What happens if the upper limit of the integral changes from \(\infty\) to a finite number?
3. Can we use a substitution method to solve this integral instead of integration by parts?
4. How can this technique be extended to integrals involving higher powers of \(x\)?
5. How does the result change if the integrand is \(xe^{-2x}\)?

**Tip:** When integrating functions of the form \( x^n e^{-x} \), using the integration by parts method multiple times is a systematic approach.

Evaluate the improper integral: ∫ from 0 to ∞ of x * e^(-x) dx.

Learn how to evaluate the improper integral ∫ from 0 to ∞ of x * e^(-x) dx using integration by parts. This problem covers key calculus concepts, including exponential functions and the integration by parts theorem, suitable for undergraduate students.

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