Worksheet: Convergence and Divergence of Improper Integrals

In this worksheet, we will practice determining the type of an improper integral over an infinite interval and evaluating the integral if it is convergent.

Q1:

Determine whether the integral ο„Έ 𝑒 πœƒ πœƒ ∞ 0 πœƒ c o s s i n d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q2:

Determine whether the integral ο„Έ ο€Ή 𝑦 βˆ’ 3 𝑦  𝑦 ∞ βˆ’ ∞ 3 2 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q3:

Determine whether the integral ο„Έ 1 3 βˆ’ 4 π‘₯ π‘₯ 0 βˆ’ ∞ d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q4:

Determine whether the integral ο„Έ 1 √ 1 + π‘₯ π‘₯ ∞ 0 4 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q5:

Determine whether the integral ο„Έ π‘₯ √ 1 + π‘₯ π‘₯ ∞ 0 2 3 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q6:

Determine whether the integral ο„Έ 𝛼 𝛼 ∞ 0 2 s i n d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q7:

Determine whether the integral ο„Έ π‘₯ π‘₯ π‘₯ ∞ 1 l n d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q8:

The integral ο„Έ 2 π‘Ÿ 0 βˆ’ ∞ π‘Ÿ d is convergent. What does it converge to?

  • A 2 2 l n
  • B βˆ’ 1 2 l n
  • C βˆ’ 2 l n
  • D 1 2 l n
  • E 2 2 l n

Q9:

The integral ο„Έ 𝑣 𝑣 + 2 𝑣 βˆ’ 3 ∞ 2 2 d is convergent. What does it converge to?

  • A 4 5 l n
  • B βˆ’ 5 4 l n
  • C βˆ’ 4 5 l n
  • D l n 5 4
  • E0

Q10:

The integral ο„Έ 1 π‘₯ + π‘₯ π‘₯ ∞ 1 2 d is convergent. What does it converge to?

  • A0
  • B βˆ’ 2 l n
  • C 1 2 l n
  • D l n 2
  • E βˆ’ 1 2 l n

Q11:

The integral ο„Έ 𝑧 𝑧 + 4 𝑧 0 βˆ’ ∞ 4 d is convergent. What does it converge to?

  • A βˆ’ πœ‹ 2
  • B πœ‹ 8
  • C πœ‹ 2
  • D βˆ’ πœ‹ 8
  • E βˆ’ πœ‹ 1 6

Q12:

The integral ο„Έ 1 ( π‘₯ βˆ’ 2 ) π‘₯ ∞ 3 3 2 d is convergent. What does it converge to?

Q13:

The integral ο„Έ π‘₯ √ π‘₯ + π‘₯ √ π‘₯ ∞ 1 d is convergent. What does it converge to?

  • A πœ‹ 4
  • B 3 πœ‹ 2
  • C 3 πœ‹ 4
  • D πœ‹ 2
  • E0

Q14:

The integral ο„Έ 𝑒 𝑝 ∞ 2 βˆ’ 5 𝑝 d is convergent. What does it converge to?

  • A 5 𝑒 βˆ’ 1 0
  • B βˆ’ 𝑒 5 βˆ’ 1 0
  • C βˆ’ 5 𝑒 βˆ’ 5
  • D 𝑒 5 βˆ’ 1 0
  • E βˆ’ 𝑒 5 βˆ’ 5

Q15:

The integral ο„Έ π‘₯ 𝑒 π‘₯ ∞ βˆ’ ∞ βˆ’ π‘₯ 2 d is convergent. What does it converge to?

Q16:

The integral ο„Έ 𝑧 𝑒 𝑧 0 βˆ’ ∞ 2 𝑧 d is convergent. What does it converge to?

  • A 3 4
  • B 1 4
  • C βˆ’ 2
  • D βˆ’ 1 4
  • E2

Q17:

The integral ο„Έ 𝑒 π‘₯ π‘₯ ∞ 1 βˆ’ 2 1 π‘₯ d is convergent. What does it converge to?

  • A 1 𝑒
  • B 1 βˆ’ 𝑒
  • C 1 𝑒 βˆ’ 1
  • D 1 βˆ’ 1 𝑒
  • E 1 + 1 𝑒

Q18:

The integral ο„Έ 𝑦 𝑒 𝑦 ∞ 2 βˆ’ 3 𝑦 d is convergent. What does it converge to?

  • A 6 3 𝑒 βˆ’ 6
  • B βˆ’ 7 𝑒 9 βˆ’ 6
  • C βˆ’ 6 3 𝑒 βˆ’ 3
  • D 7 𝑒 9 βˆ’ 6
  • E 7 𝑒 9 βˆ’ 3

Q19:

The integral ο„Έ 𝑒 𝑦 ∞ 0 βˆ’ √ 𝑦 d is convergent. What does it converge to?

Q20:

The integral ο„Έ π‘₯ π‘₯ π‘₯ ∞ 1 2 l n d is convergent. What does it converge to?

Q21:

Consider the integral ο„Έ 1 ( 2 π‘₯ + 1 ) π‘₯ ∞   d .

Determine whether the integral is convergent or divergent.

  • AConvergent
  • BDivergent

Determine the value of integration.

  • A 1 3 6
  • B βˆ’ 2 9
  • C 5 1 8
  • D 1 4
  • E βˆ’ 1 3 6

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