Worksheet: Improper Integrals: Infinite Limits of Integration

In this worksheet, we will practice evaluating improper integrals where one or more of the endpoints approach infinity.

Q1:

Is ο„Έ 𝑒 π‘₯ ∞ 0 βˆ’ π‘₯ 3 d an improper integral?

  • Ayes
  • Bno

Q2:

Consider the integral ο„Έ ∞  1 π‘₯ ( l n π‘₯ )  d π‘₯ .

Find all possible values of 𝑝 for which the integral is convergent.

  • A 𝑝 < 1
  • B 𝑝 ≀ 1
  • C 𝑝 β‰₯ 1
  • D 𝑝 > 1
  • E 𝑝 = 1

Evaluate the integral for those values of 𝑝 .

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

Q3:

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

Q4:

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

Q5:

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

Q6:

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

  • Adivergent
  • Bconvergent

Q7:

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

Q8:

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

Q9:

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

  • Adivergent
  • Bconvergent

Q10:

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

Q11:

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

Q12:

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

Q13:

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

  • Adivergent
  • Bconvergent

Q14:

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

Q15:

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

  • Adivergent
  • Bconvergent

Q16:

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

Q17:

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

Q18:

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

Q19:

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

Q20:

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

Q21:

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

  • Adivergent
  • Bconvergent

Q22:

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

Q23:

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

Q24:

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

  • Adivergent
  • Bconvergent

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