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?

  • Aβˆ’14
  • Bβˆ’12
  • C14
  • D12
  • E0

Q2:

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

  • ADivergent
  • BConvergent

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?

  • Aβˆ’πœ‹2
  • Bπœ‹4
  • C0
  • Dβˆ’πœ‹4
  • Eπœ‹2

Q5:

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

  • Aβˆ’345
  • B235
  • C345
  • Dβˆ’325
  • E325

Q6:

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

  • A6
  • Bβˆ’32
  • C323
  • Dβˆ’323
  • Eβˆ’2

Q7:

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

  • Aconvergent
  • Bdivergent

Q8:

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

  • A2𝑒
  • Bβˆ’1𝑒
  • Cβˆ’2𝑒
  • Dβˆ’2𝑒
  • E𝑒

Q9:

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

  • Adivergent
  • Bconvergent

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.

  • Aconvergent
  • Bdivergent

Q13:

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

  • Adivergent
  • Bconvergent

Q14:

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

  • A92
  • Bβˆ’12
  • Cβˆ’152
  • Dβˆ’1
  • E2

Q15:

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

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

  • A𝑝=βˆ’1
  • B𝑝>βˆ’1
  • C𝑝β‰₯βˆ’1
  • D𝑝<βˆ’1
  • Eπ‘β‰€βˆ’1

Evaluate the integral for those values of 𝑝.

  • Aβˆ’1π‘βˆ’1
  • Bβˆ’1(𝑝+1)
  • Cβˆ’1(π‘βˆ’1)
  • D1𝑝+1
  • E1(𝑝+1)

Q16:

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

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

  • A𝑝>1
  • B𝑝⩾1
  • C𝑝⩽1
  • D𝑝=1
  • E𝑝<1

Evaluate the integral for those values of 𝑝.

  • A1𝑝
  • B1βˆ’π‘
  • C11βˆ’π‘
  • D1+𝑝
  • E11+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 undefined if π‘Ž<0, 𝑏>0 and |π‘Ž|β‰ |𝑏|.
  • CThe integral is defined for all finite values of π‘Ž and 𝑏.
  • DThe integral is undefined only if π‘Ž=0 or 𝑏=0.
  • EThe integral is undefined if π‘Ž<0 and 𝑏>0.

Q18:

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

  • Ayes
  • Bno

Q19:

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

  • AYes
  • BNo

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.