Worksheet: Improper Integrals: Discontinuous Integrands

In this worksheet, we will practice evaluating improper integrals where the integrand has a vertical asymptote.

Q1:

The integral ο„Έπ‘Ÿπ‘Ÿπ‘ŸοŠ§οŠ¦lnd is convergent. What does it converge to?

  • A 1 4
  • B0
  • C βˆ’ 1 2
  • D 1 2
  • E βˆ’ 1 4

Q2:

Determine whether the integral ο„Έ1π‘₯π‘₯d is convergent or divergent.

  • AConvergent
  • BDivergent

Q3:

Determine whether the integral ο„Έπ‘₯(π‘₯+1)π‘₯d is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q4:

The integral ο„Έ1√1βˆ’π‘₯π‘₯d is convergent. What does it converge to?

  • A0
  • B βˆ’ πœ‹ 2
  • C πœ‹ 4
  • D βˆ’ πœ‹ 4
  • E πœ‹ 2

Q5:

The integral ο„Έ1√5βˆ’π‘₯π‘₯d is convergent. What does it converge to?

  • A 3 4 5   
  • B βˆ’ 3 2 5  
  • C 3 2 5  
  • D 2 3 5  
  • E βˆ’ 3 4 5   

Q6:

The integral ο„Έ1√π‘₯+2π‘₯οŠͺd is convergent. What does it converge to?

  • A6
  • B 3 2 3
  • C βˆ’ 3 2 3
  • D βˆ’ 3 2
  • E βˆ’ 2

Q7:

Determine whether the integral 𝑒π‘₯π‘₯οŠ§οŠ¦οŠ©οŽ ο‘d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q8:

The integral 𝑒π‘₯π‘₯οŠ¦οŠ±οŠ§οŠ©οŽ ο‘d is convergent. What does it converge to?

  • A 2 𝑒
  • B βˆ’ 2 𝑒
  • C βˆ’ 1 𝑒
  • D 𝑒
  • E βˆ’ 2 𝑒

Q9:

Determine whether the integral ο„Έπœƒπœƒο‘½οŽ‘οŠ¦οŠ¨tand is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q10:

The integral ο„Έπœƒβˆšπœƒπœƒο‘½οŽ‘οŠ¦cossind is convergent. What does it converge to?

Q11:

Determine whether the integral ο„Έ1π‘₯π‘₯οŠͺd is convergent or divergent.

  • Adivergent
  • Bconvergent

Q12:

Determine whether the integral ο„Έπ‘€π‘€βˆ’2π‘€οŠ«οŠ¦d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q13:

Determine whether the integral ο„Έ1π‘₯βˆ’π‘₯βˆ’2π‘₯οŠͺd is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q14:

The integral ο„Έ1√π‘₯βˆ’1π‘₯d is convergent. What does it converge to?

  • A βˆ’ 1
  • B βˆ’ 1 2
  • C2
  • D 9 2
  • E βˆ’ 1 5 2

Q15:

Consider the integral ο„Έπ‘₯π‘₯π‘₯lnd.

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

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

Evaluate the integral for those values of 𝑝.

  • A βˆ’ 1 𝑝 βˆ’ 1
  • B 1 𝑝 + 1
  • C βˆ’ 1 ( 𝑝 βˆ’ 1 ) 
  • D 1 ( 𝑝 + 1 ) 
  • E βˆ’ 1 ( 𝑝 + 1 ) 

Q16:

Consider the integral ο„Έ1π‘₯π‘₯d.

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

  • A 𝑝 β©½ 1
  • B 𝑝 β©Ύ 1
  • C 𝑝 = 1
  • D 𝑝 < 1
  • E 𝑝 > 1

Evaluate the integral for those values of 𝑝.

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

Q17:

Consider the integral ο„Έπ‘’βˆ’π‘’π‘’+𝑒π‘₯οŒ»οŒΊο—οŠ±ο—ο—οŠ±ο—d. Which of the following is a correct characterization of this integral?

  • AThe integral is undefined only if π‘Ž or 𝑏 is an integer multiple of πœ‹.
  • BThe integral is defined for all finite values of π‘Ž and 𝑏.
  • CThe integral is undefined if π‘Ž<0 and 𝑏>0.
  • DThe integral is undefined if π‘Ž<0, 𝑏>0 and |π‘Ž|β‰ |𝑏|.
  • EThe integral is undefined only if π‘Ž=0 or 𝑏=0.

Q18:

Is ο„Έπ‘₯π‘₯ο‘½οŽ£οŠ¦tand an improper integral?

  • Ano
  • Byes

Q19:

Is ο„Έπ‘₯π‘₯οŽ„οŠ¦tand an improper integral?

  • AYes
  • BNo

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