To solve the integral \( \int_{0}^{\infty} e^{-kx} \, dx \), where \( k \) is a positive constant, we can follow similar steps to those used in the previous problem.

### Steps:

1. **Find the antiderivative**:
   The antiderivative of \( e^{-kx} \) with respect to \( x \) is:

\[
\int e^{-kx} \, dx = -\frac{1}{k} e^{-kx} + C
\]

2. **Evaluate the definite integral**:
   We now evaluate this from \( x = 0 \) to \( x = \infty \):

\[
\int_{0}^{\infty} e^{-kx} \, dx = \left[-\frac{1}{k} e^{-kx} \right]_{0}^{\infty}
\]

3. **Compute the limits**:

   - As \( x \) approaches infinity, \( e^{-kx} \) approaches 0 (since \( k > 0 \)).
   - At \( x = 0 \), \( e^{-kx} = e^{0} = 1 \).

\[
\left( -\frac{1}{k} e^{-k\infty} \right) - \left( -\frac{1}{k} e^{0} \right) = 0 - \left(-\frac{1}{k} \right) = \frac{1}{k}
\]

Thus, the integral evaluates to:

\[
\int_{0}^{\infty} e^{-kx} \, dx = \frac{1}{k}
\]

Would you like more details or have any questions?

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Here are 8 related questions that expand on this topic:

1. How does the value of \( k \) affect the rate of decay in the function \( e^{-kx} \)?
2. What is the physical interpretation of this integral in terms of exponential decay processes?
3. How would you compute the integral if \( k \) were a negative constant?
4. Can you evaluate \( \int_{a}^{b} e^{-kx} \, dx \) for arbitrary limits \( a \) and \( b \)?
5. What is the relationship between this integral and the Laplace transform of 1?
6. How does this integral relate to the concept of a mean lifetime in physics?
7. What is the general form of the integral \( \int_{0}^{\infty} x^n e^{-kx} \, dx \) for a non-negative integer \( n \)?
8. Can you derive this result using a substitution method instead of direct integration?

**Tip:** Integrals involving exponential functions often appear in applications such as probability, physics, and engineering, making it essential to understand their evaluation and interpretation.

Learn how to evaluate the integral ∫₀ᴵⁿᶠᶦⁿᶦᵗʸ e⁻ᵏˣ dx step-by-step with a detailed solution. This problem involves the antiderivative of exponential functions and its evaluation for positive constants. Ideal for undergraduate students studying integral calculus and exponential decay processes.

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