integrate between 4 and infiniti the function xe^(-4x)dx

We are asked to evaluate the improper integral:

\[
\int_{4}^{\infty} x e^{-4x} \, dx
\]

### Step 1: Recognize the integral form
The integral involves the function \( x e^{-4x} \), which suggests integration by parts.

### Step 2: Apply Integration by Parts
Using the integration by parts formula:

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

Let:
- \( u = x \), so \( du = dx \)
- \( dv = e^{-4x} \, dx \), so \( v = \int e^{-4x} \, dx = \frac{-1}{4} e^{-4x} \)

Now apply the formula:

\[
\int x e^{-4x} \, dx = x \left( \frac{-1}{4} e^{-4x} \right) - \int \frac{-1}{4} e^{-4x} \, dx
\]

### Step 3: Simplify the terms
This simplifies to:

\[
= \frac{-x}{4} e^{-4x} + \frac{1}{16} e^{-4x}
\]

### Step 4: Evaluate the definite integral
Now, evaluate the expression from \( 4 \) to \( \infty \):

\[
\int_{4}^{\infty} x e^{-4x} \, dx = \left[ \frac{-x}{4} e^{-4x} + \frac{1}{16} e^{-4x} \right]_{4}^{\infty}
\]

First, evaluate the limit as \( x \to \infty \). Since \( e^{-4x} \) decays to zero very quickly as \( x \) increases:

\[
\lim_{x \to \infty} \left( \frac{-x}{4} e^{-4x} + \frac{1}{16} e^{-4x} \right) = 0
\]

Now, substitute \( x = 4 \):

\[
\left( \frac{-4}{4} e^{-16} + \frac{1}{16} e^{-16} \right) = \left( -e^{-16} + \frac{1}{16} e^{-16} \right) = -\frac{15}{16} e^{-16}
\]

Thus, the value of the integral is:

\[
\int_{4}^{\infty} x e^{-4x} \, dx = \frac{15}{16} e^{-16}
\]

### Final Answer:
\[
\frac{15}{16} e^{-16}
\]

Would you like more details or have any questions?

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Here are 5 related questions to explore further:
1. How do you solve an improper integral when the upper limit is infinity?
2. Can integration by parts be used to solve all exponential functions?
3. What happens if the exponent in the exponential function changes?
4. How do you handle integrals that involve products of polynomials and exponentials?
5. How does the decay rate of \( e^{-4x} \) affect the convergence of this integral?

**Tip:** When evaluating limits at infinity in integrals, remember that exponential decay (like \( e^{-4x} \)) will often force terms to zero very quickly.

integrate between 4 and infinity the function xe^(-4x)dx

This solution demonstrates how to evaluate the improper integral of xe^(-4x) from 4 to infinity using integration by parts. The step-by-step guide covers key calculus concepts, including integration by parts and handling limits at infinity. Suitable for college-level students studying calculus.

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