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Chapter 6: Problem 3

Determine whether or not the integral is improper. $$\int_{0}^{2} x^{2 / 5} d x$$

Short Answer

Expert verified
No, the integral \(\int_{0}^{2} x^{2 / 5} dx\) is not improper.

Step by step solution

01

Inspect the Limits of Integration

The limits of the given integral \(\int_{0}^{2} x^{2 / 5} dx\) are 0 and 2. Both are particular, finite numbers.
02

Examine the function

The function to be integrated is \(x^{2/5}\). This function is a standard polynomial and is continuous on all of its domain, including specifically on the interval [0, 2] under consideration.
03

Conclusion

Since neither of the limits of integration is infinite and the function is continuous within the interval [0, 2], the given integral is not improper.

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Key Concepts

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

Limits of Integration
When evaluating an integral like \( \int_{0}^{2} x^{2 / 5} dx \), the limits of integration are crucial to determine whether the integral is proper or improper. The limits indicate the range over which the integration occurs.
In this exercise, the limits are 0 and 2, which are both finite values. If either limit was infinite, or if the integral involved a point where the function is undefined, then it would be considered improper.
  • Finite Limits: When both limits are finite and the function within these limits is well-defined, we are safely dealing with a proper integral.
  • Infinite Limits: If one or both limits extend to infinity, additional steps are required to evaluate the improper integral, often using limits.
  • Undefined Points: If the function has points within the limits where it is not defined (like where the function might become infinite), this too might cause an integral to be classified as improper.
Polynomial Functions
The given function \( x^{2/5} \) is an example of a polynomial function. Simply put, polynomial functions are expressions involving variables raised to a power and multiplied by coefficients.
They are straightforward to work with because they follow predictable patterns.
  • Structure: A polynomial鈥檚 structure is based on terms of the form \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer or a rational number as in this case.
  • Operations: Integrating polynomial functions typically involves applying the power rule, which simplifies the process significantly.
  • Solutions: With functions like \( x^{2/5} \), recognizing them as polynomial aids in predicting their behavior over intervals.
In the context of integration, our function \( x^{2/5} \) is raised to a non-integer power, but it is still treated similarly for integration purposes.
Continuous Functions
Understanding continuity is key when assessing whether a function, like \( x^{2/5} \), is fit to be integrated over a given interval. A continuous function has no breaks, jumps, or points where it goes to infinity within its domain.
  • Characteristics: Continuous functions ensure that you can draw them without lifting your pen.
  • Importance in Integration: For integrals, continuity assures that the function behaves predictably, allowing you to calculate the area under the curve smoothly.
  • Continuous Domain: In the interval from 0 to 2, \( x^{2/5} \) remains continuous, meaning it has no undefined points or discontinuities.
Checking continuity is a step you can't skip, as it reassures that the integral is properly defined over the interval, with no unexpected behavior from the function.

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Most popular questions from this chapter

Given that \(\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi},\) evaluate \(\int_{-\infty}^{\infty} e^{-k x^{2}} d x\) for \(k > 0\)

In exercises find the partial fractions decomposition and an antiderivative. If you have a CAS available, use it to check your answer. $$\frac{3 x+8}{x^{3}+5 x^{2}+6 x}$$

Find out if your CAS has a special command (e.g., APART in Mathematica) to do partial fractions decompositions. Also, try \(\int \frac{x^{2}+2 x-1}{(x-1)^{2}\left(x^{2}+4\right)} d x\) and \, \(\int \frac{3 x}{\left(x^{2}+x+2\right)^{2}} d x.\)

Use a comparison to determine whether the integral converges or diverges. $$\int_{2}^{\infty} \frac{\ln x}{e^{x}+1} d x$$

Show that \(\int_{-\infty}^{\infty} x^{p} d x\) diverges for every \(p\)
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