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Chapter 5: Problem 15

Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent. \(\int_{-\infty}^{\infty} 2 x e^{-3 x^{2}} d x\)

Short Answer

Expert verified
The integral converges to 0.

Step by step solution

01

- Recognize the Improper Integral

Given the integral \(\int_{-\infty}^{\infty} 2 x e^{-3 x^{2}} d x\), it is an improper integral because the limits of integration are infinite.
02

- Break the Integral into Two Parts

Split the integral at 0: \[ \int_{-\infty}^{\frac{0}} 2 x e^{-3 x^{2}} d x + \int_{0}^{\infty} 2 x e^{-3 x^{2}} d x \]
03

- Use Symmetry of the Function

Notice that the integrand function \(2 x e^{-3x^{2}}\) is an odd function because \(f(-x) = -f(x)\). The integral of an odd function over symmetric limits \(\int_{-a}^a f(x) dx\) is zero. Therefore, \[ \int_{-\infty}^{\infty} 2 x e^{-3 x^{2}} d x= 0 \]
04

Conclusion

The improper integral converges to 0 because the function is odd and symmetric about the y-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence and Divergence
When dealing with improper integrals, we first need to check if the integral converges or diverges. Convergence means the integral has a finite value, while divergence means it does not.
For an integral like \(\backslash\text{鈭珆_{-\backslash\text{infty}}^{\backslash\text{infty}} 2 x e^{-3 x^{2}} d x\), we are integrating over an infinite range. To determine convergence, examine if the integral's value approaches a finite number as the limits go to infinity.
In our case, by splitting the integral at zero, we investigate the behavior toward positive and negative infinity separately before concluding the total integral's convergence.
Odd Functions
An odd function is a function where \(\backslash\text{f}(-x) = -f(x)\). This symmetry is crucial in integration. In our problem, the integrand \(2 x e^{-3 x^{2}}\) is an odd function.
Recognizing this helps simplify our work. If you integrate an odd function over symmetric limits, such as from \(-a\) to \(a\), the result is always zero. This is because the negative part of the function cancels the positive part.
Knowing the function is odd allowed us to find the integral's value quickly, providing a clear path to determine whether the integral converges.
Symmetry in Integration
Symmetry can greatly simplify the integration process. In our exercise, we split the integral at 0, capitalizing on the function's symmetry.
Once we found that \(2 x e^{-3 x^{2}}\) is an odd function, we used symmetry. Integrating from \(-\backslash\text{infty}\) to 0 cancels the integral from 0 to \(backslash\text{infty}\).
This approach makes it easier to recognize and apply properties of the function. By using the function's symmetry, we concluded the integral over the entire domain.
Infinite Limits
Integrating with infinite limits involves evaluating the behavior of the function as it approaches infinity. With \(\backslash\text{鈭珆_{-\backslash\text{infty}}^{\backslash\text{infty}} 2 x e^{-3 x^{2}} d x\), we deal with both positive and negative infinity.
Splitting the integral allows examination of the bounds separately. First, look at how the function behaves towards infinity and negative infinity.
This is where the concepts of convergence and divergence apply again. By ensuring each part of the integral is defined and behaves properly, we confirm if the entire improper integral converges and, if so, find its value.

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Most popular questions from this chapter

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