Evaluate The Following Improper Integral. If The Integral Diverges, Enter DIV.${ \int_1^{\infty} X E^{-x} , Dx = , \square }$

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Introduction

Improper integrals are a crucial concept in calculus, allowing us to evaluate integrals that have infinite limits of integration or integrands that become infinite at certain points. In this article, we will focus on evaluating the improper integral ∫1∞xe−x dx\int_1^{\infty} x e^{-x} \, dx. We will explore the concept of improper integrals, discuss the conditions for convergence, and provide a step-by-step solution to the given integral.

What are Improper Integrals?

An improper integral is an integral that has infinite limits of integration or an integrand that becomes infinite at certain points. In the case of the given integral, we have an infinite upper limit of integration, which makes it an improper integral.

Conditions for Convergence

For an improper integral to converge, the following conditions must be met:

  1. The integrand must be continuous: The function being integrated must be continuous at all points in the interval of integration.
  2. The limits of integration must be finite: The lower and upper limits of integration must be finite numbers.
  3. The integrand must not become infinite: The function being integrated must not become infinite at any point in the interval of integration.

Evaluating the Improper Integral

To evaluate the improper integral ∫1∞xe−x dx\int_1^{\infty} x e^{-x} \, dx, we can use integration by parts. Integration by parts is a technique used to integrate products of functions.

Step 1: Integration by Parts

Let u=xu = x and dv=e−xdxdv = e^{-x} dx. Then, du=dxdu = dx and v=−e−xv = -e^{-x}.

Step 2: Apply the Integration by Parts Formula

The integration by parts formula is given by:

∫udv=uv−∫vdu\int u dv = uv - \int v du

Substituting the values of uu, vv, dudu, and dvdv, we get:

∫xe−xdx=−xe−x−∫−e−xdx\int x e^{-x} dx = -x e^{-x} - \int -e^{-x} dx

Step 3: Evaluate the Integral

Evaluating the integral on the right-hand side, we get:

∫xe−xdx=−xe−x+∫e−xdx\int x e^{-x} dx = -x e^{-x} + \int e^{-x} dx

The integral on the right-hand side is a standard integral, which can be evaluated as:

∫e−xdx=−e−x\int e^{-x} dx = -e^{-x}

Step 4: Apply the Limits of Integration

Applying the limits of integration, we get:

∫1∞xe−xdx=lim⁡b→∞[−xe−x−e−x]1b\int_1^{\infty} x e^{-x} dx = \lim_{b \to \infty} \left[ -x e^{-x} - e^{-x} \right]_1^b

Step 5: Evaluate the Limit

Evaluating the limit, we get:

lim⁡b→∞[−xe−x−e−x]1b=lim⁡b→∞[−be−b−e−b+e−1+1]\lim_{b \to \infty} \left[ -x e^{-x} - e^{-x} \right]_1^b = \lim_{b \to \infty} \left[ -b e^{-b} - e^{-b} + e^{-1} + 1 \right]

Step 6: Simplify the Expression

Simplifying the expression, we get:

lim⁡b→∞[−be−b−e−b+e−1+1]=1+e−1\lim_{b \to \infty} \left[ -b e^{-b} - e^{-b} + e^{-1} + 1 \right] = 1 + e^{-1}

Conclusion

In conclusion, the improper integral ∫1∞xe−x dx\int_1^{\infty} x e^{-x} \, dx converges to 1+e−11 + e^{-1}.

Discussion

Improper integrals are a crucial concept in calculus, allowing us to evaluate integrals that have infinite limits of integration or integrands that become infinite at certain points. In this article, we evaluated the improper integral ∫1∞xe−x dx\int_1^{\infty} x e^{-x} \, dx using integration by parts. We discussed the conditions for convergence and provided a step-by-step solution to the given integral.

References

  • [1] Calculus by Michael Spivak
  • [2] Calculus: Early Transcendentals by James Stewart

Additional Resources

  • [1] Khan Academy: Improper Integrals
  • [2] MIT OpenCourseWare: Calculus

Final Answer

Introduction

Improper integrals are a fundamental concept in calculus, allowing us to evaluate integrals that have infinite limits of integration or integrands that become infinite at certain points. In our previous article, we evaluated the improper integral ∫1∞xe−x dx\int_1^{\infty} x e^{-x} \, dx using integration by parts. In this article, we will provide a Q&A guide to help you better understand improper integrals.

Q: What is an improper integral?

A: An improper integral is an integral that has infinite limits of integration or an integrand that becomes infinite at certain points.

Q: What are the conditions for convergence of an improper integral?

A: For an improper integral to converge, the following conditions must be met:

  1. The integrand must be continuous: The function being integrated must be continuous at all points in the interval of integration.
  2. The limits of integration must be finite: The lower and upper limits of integration must be finite numbers.
  3. The integrand must not become infinite: The function being integrated must not become infinite at any point in the interval of integration.

Q: How do I evaluate an improper integral?

A: To evaluate an improper integral, you can use integration by parts, substitution, or other techniques. The choice of technique depends on the specific integral and the limits of integration.

Q: What is integration by parts?

A: Integration by parts is a technique used to integrate products of functions. It involves choosing two functions, u and v, and then applying the formula:

∫udv=uv−∫vdu\int u dv = uv - \int v du

Q: How do I choose u and v for integration by parts?

A: To choose u and v, you can use the following guidelines:

  • Choose u to be the function that becomes simpler after differentiation.
  • Choose dv to be the function that becomes simpler after integration.

Q: What is the difference between a proper integral and an improper integral?

A: A proper integral is an integral with finite limits of integration, while an improper integral is an integral with infinite limits of integration or an integrand that becomes infinite at certain points.

Q: Can an improper integral have a finite value?

A: Yes, an improper integral can have a finite value. This occurs when the integrand becomes infinite at a single point, but the integral converges to a finite value.

Q: How do I determine if an improper integral converges or diverges?

A: To determine if an improper integral converges or diverges, you can use the following guidelines:

  • If the integrand becomes infinite at a single point, but the integral converges to a finite value, then the integral converges.
  • If the integrand becomes infinite at multiple points, or if the integral diverges to infinity, then the integral diverges.

Q: What are some common improper integrals?

A: Some common improper integrals include:

  • ∫1∞1x2dx\int_1^{\infty} \frac{1}{x^2} dx
  • ∫0∞e−xdx\int_0^{\infty} e^{-x} dx
  • ∫1∞xe−xdx\int_1^{\infty} x e^{-x} dx

Conclusion

In conclusion, improper integrals are a fundamental concept in calculus, allowing us to evaluate integrals that have infinite limits of integration or integrands that become infinite at certain points. By understanding the conditions for convergence, choosing the right technique, and following the guidelines for integration by parts, you can evaluate improper integrals with confidence.

References

  • [1] Calculus by Michael Spivak
  • [2] Calculus: Early Transcendentals by James Stewart

Additional Resources

  • [1] Khan Academy: Improper Integrals
  • [2] MIT OpenCourseWare: Calculus

Final Answer

The final answer is: 1+e−1\boxed{1 + e^{-1}}