The integral of \( e^{-x} \) from 0 to infinity is a well-known improper integral in calculus. Let's compute it:

\[
\int_{0}^{\infty} e^{-x} \, dx
\]

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

\[
\int e^{-x} \, dx = -e^{-x} + C
\]

2. **Evaluate the definite integral**:
   Now, we evaluate the definite integral from 0 to infinity:

\[
\left[ -e^{-x} \right]_{0}^{\infty}
\]

3. **Compute the limits**:

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

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

Thus, the integral evaluates to:

\[
\int_{0}^{\infty} e^{-x} \, dx = 1
\]

Would you like more details or have any questions?

---

Here are 8 related questions that expand on this topic:

1. What is the significance of improper integrals in calculus?
2. How do you evaluate integrals involving exponential functions with different bases?
3. Can you explain the concept of convergence in the context of improper integrals?
4. What is the integral of \( e^{-kx} \) from 0 to infinity where \( k \) is a positive constant?
5. How does the integral of \( e^{-x^2} \) differ from \( e^{-x} \)?
6. What is the relationship between this integral and the Laplace transform?
7. Can you calculate the definite integral of \( e^{-x} \) from \( a \) to \( b \)?
8. How would you solve this integral using numerical methods?

**Tip:** When evaluating improper integrals, always check for convergence by analyzing the behavior of the integrand at the boundaries.

Learn how to evaluate the integral of e^{-x} from 0 to infinity step-by-step. Understand the concept of improper integrals in calculus, compute the antiderivative of e^{-x}, and apply convergence criteria. Suitable for undergraduate students and anyone interested in calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation