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Lesson Worksheet: Comparison Test for Improper Integrals Mathematics

In this worksheet, we will practice determining whether an improper integral is convergent or divergent using the comparison test for improper integrals.

Q1:

Use the comparison theorem to determine whether the integral ο„Έπ‘₯π‘₯+1π‘₯∞d is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q2:

Use the comparison theorem to determine whether the integral ο„Έπ‘₯+1√π‘₯βˆ’π‘₯π‘₯∞οŠͺd is convergent or divergent.

  • Adivergent
  • Bconvergent

Q3:

Use the comparison theorem to determine whether the integral ο„Έ1+π‘₯√π‘₯π‘₯∞sind is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q4:

Use the comparison theorem to determine whether the integral ο„Έπ‘₯√π‘₯π‘₯οŽ„οŠ¦οŠ¨sind is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q5:

Use the comparison theorem to determine whether the integral ο„Έπ‘₯π‘₯√π‘₯π‘₯secd is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q6:

Use the comparison theorem to determine whether the integral ο„Έπ‘₯2+𝑒π‘₯βˆžοŠ¦οŠ±οŠ§ο—tand is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q7:

Determine whether ο„Έπ‘₯π‘₯π‘₯∞arctand is convergent, divergent, or cannot be determined.

  • ADivergent
  • BCannot be determined
  • CConvergent

Q8:

For what values of 𝑝 is the integral ο„Έ1π‘₯π‘₯∞d convergent?

  • A𝑝β‰₯1
  • B0<𝑝<1
  • C𝑝>1
  • D𝑝<1
  • E0<𝑝≀1

Q9:

For what values of 𝑝 is the integral ο„Έ1π‘₯π‘₯d convergent?

  • A0<𝑝<1
  • B𝑝β‰₯1
  • C𝑝<1
  • D𝑝>1
  • E𝑝≀1

Q10:

Suppose that 𝑓 and 𝑔 are continuous and 0≀𝑔(π‘₯)≀𝑓(π‘₯) for all π‘₯β‰₯π‘Ž.

Which of the following statements is true?

  • AIf 𝑔(π‘₯)π‘₯∞d is convergent, then 𝑓(π‘₯)π‘₯∞d is convergent, and if 𝑔(π‘₯)π‘₯∞d is divergent, then 𝑓(π‘₯)π‘₯∞d is divergent.
  • BIf 𝑓(π‘₯)π‘₯∞d is convergent, then 𝑔(π‘₯)π‘₯∞d is convergent, and if 𝑓(π‘₯)π‘₯∞d is divergent, then 𝑔(π‘₯)π‘₯∞d is divergent.
  • CIf 𝑓(π‘₯)π‘₯∞d is convergent, then 𝑔(π‘₯)π‘₯∞d is convergent, and if 𝑔(π‘₯)π‘₯∞d is divergent, then 𝑓(π‘₯)π‘₯∞d is divergent.
  • DIf 𝑓(π‘₯)π‘₯∞d is convergent, then 𝑔(π‘₯)π‘₯∞d is divergent, and if 𝑔(π‘₯)π‘₯∞d is divergent, then 𝑓(π‘₯)π‘₯∞d is divergent.
  • EIf 𝑓(π‘₯)π‘₯∞d is divergent, then 𝑔(π‘₯)π‘₯∞d is convergent, and if 𝑔(π‘₯)π‘₯∞d is divergent, then 𝑓(π‘₯)π‘₯∞d is divergent.

This lesson includes 9 additional question variations for subscribers.

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