The integral you provided is:

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

This is a standard integral in calculus and can be solved using integration by parts. Let's solve it step by step:

### Step 1: Integration by Parts Formula

The formula for integration by parts is:

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

We need to choose \( u \) and \( dv \) appropriately. Let’s set:

\[
u = x \quad \text{and} \quad dv = e^{-x} dx
\]

### Step 2: Differentiate and Integrate

- Differentiate \( u = x \) to get \( du = dx \).
- Integrate \( dv = e^{-x} dx \) to get \( v = -e^{-x} \).

### Step 3: Apply the Integration by Parts Formula

Now, apply the formula:

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

### Step 4: Evaluate the Boundary Terms

- As \( x \to \infty \), \( -x e^{-x} \to 0 \) because the exponential decay is faster than the linear term.
- At \( x = 0 \), \( -x e^{-x} = 0 \).

Thus, the boundary term evaluates to 0.

### Step 5: Solve the Remaining Integral

We now have:

\[
\int_0^\infty x e^{-x} \, dx = 0 + \int_0^\infty e^{-x} \, dx
\]

The remaining integral is:

\[
\int_0^\infty e^{-x} \, dx = 1
\]

### Final Answer:

Thus, the value of the integral is:

\[
\int_0^\infty x e^{-x} \, dx = 1
\]

Would you like more details on this solution?

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Here are five related questions to explore:

1. How does integration by parts work with more complex functions?
2. What is the general form of integrals involving \( x^n e^{-x} \)?
3. How do you evaluate integrals from 0 to infinity with trigonometric functions?
4. Can you apply substitution methods to solve this type of integral?
5. How can Laplace transforms be related to integrals like these?

**Tip:** When solving integrals involving exponentials, integration by parts is often useful, especially when there's a product of a polynomial and an exponential function.

This tutorial shows how to solve the integral ∫(0 to ∞) x e^(-x) dx using integration by parts. Learn key calculus concepts, including improper integrals and exponential decay functions, with a step-by-step solution. Suitable for advanced high school and college students.

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