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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 𝑥𝑥+1𝑥d is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q2:

Use the comparison theorem to determine whether the integral 𝑥+1𝑥𝑥𝑥d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q3:

Use the comparison theorem to determine whether the integral 1+𝑥𝑥𝑥sind is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q4:

Use the comparison theorem to determine whether the integral 𝑥𝑥𝑥sind is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q5:

Use the comparison theorem to determine whether the integral 𝑥𝑥𝑥𝑥secd is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q6:

Use the comparison theorem to determine whether the integral 𝑥2+𝑒𝑥tand is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q7:

Determine whether 𝑥𝑥𝑥arctand is convergent, divergent, or cannot be determined.

  • ADivergent
  • BCannot be determined
  • CConvergent

Q8:

For what values of 𝑝 is the integral 1𝑥𝑥d convergent?

  • A𝑝1
  • B0<𝑝<1
  • C𝑝>1
  • D𝑝<1
  • E0<𝑝1

Q9:

For what values of 𝑝 is the integral 1𝑥𝑥d convergent?

  • A0<𝑝<1
  • B𝑝1
  • C𝑝<1
  • D𝑝>1
  • E𝑝1

Q10:

Suppose that 𝑓 and 𝑔 are continuous and 0𝑔(𝑥)𝑓(𝑥) for all 𝑥𝑎.

Which of the following statements is true?

  • AIf 𝑔(𝑥)𝑥d is convergent, then 𝑓(𝑥)𝑥d is convergent, and if 𝑔(𝑥)𝑥d is divergent, then 𝑓(𝑥)𝑥d is divergent.
  • BIf 𝑓(𝑥)𝑥d is convergent, then 𝑔(𝑥)𝑥d is convergent, and if 𝑓(𝑥)𝑥d is divergent, then 𝑔(𝑥)𝑥d is divergent.
  • CIf 𝑓(𝑥)𝑥d is convergent, then 𝑔(𝑥)𝑥d is convergent, and if 𝑔(𝑥)𝑥d is divergent, then 𝑓(𝑥)𝑥d is divergent.
  • DIf 𝑓(𝑥)𝑥d is convergent, then 𝑔(𝑥)𝑥d is divergent, and if 𝑔(𝑥)𝑥d is divergent, then 𝑓(𝑥)𝑥d is divergent.
  • EIf 𝑓(𝑥)𝑥d is divergent, then 𝑔(𝑥)𝑥d is convergent, and if 𝑔(𝑥)𝑥d is divergent, then 𝑓(𝑥)𝑥d is divergent.

This lesson includes 9 additional question variations for subscribers.

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