Worksheet: Improper Integrals with Discontinuous Integrands

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In this worksheet, we will practice determining the type of the improper integral with a discontinuous integrand and evaluating the integral if it is convergent.

Q1:

The integral ο„Έ π‘Ÿ π‘Ÿ π‘Ÿ   l n d is convergent. What does it converge to?

  • A βˆ’ 1 2
  • B 1 4
  • C0
  • D βˆ’ 1 4
  • E 1 2

Q2:

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

  • ADivergent
  • BConvergent

Q3:

Determine whether the integral ο„Έ π‘₯ ( π‘₯ + 1 ) π‘₯ 2 βˆ’ 1 2 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q4:

The integral ο„Έ 1 √ 1 βˆ’ π‘₯ π‘₯ 1 0 2 d is convergent. What does it converge to?

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

Q5:

The integral ο„Έ 1 √ 5 βˆ’ π‘₯ π‘₯ 5 0 3 d is convergent. What does it converge to?

  • A βˆ’ 3 2 5 2 3
  • B 3 4 5 βˆ’ 4 3
  • C βˆ’ 3 4 5 βˆ’ 4 3
  • D 3 2 5 2 3
  • E 2 3 5 2 3

Q6:

The integral ο„Έ 1 √ π‘₯ + 2 π‘₯ 1 4 βˆ’ 2 4 d is convergent. What does it converge to?

  • A βˆ’ 3 2
  • B6
  • C βˆ’ 2
  • D 3 2 3
  • E βˆ’ 3 2 3

Q7:

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

  • Adivergent
  • Bconvergent

Q8:

The integral ο„Έ 𝑒 π‘₯ π‘₯ 0 βˆ’ 1 3 1 π‘₯ d is convergent. What does it converge to?

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

Q9:

Determine whether the integral ο„Έ πœƒ πœƒ πœ‹ 2 0 2 t a n d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q10:

The integral ο„Έ πœƒ √ πœƒ πœƒ πœ‹ 2 0 c o s s i n d is convergent. What does it converge to?

Q11:

Determine whether the integral ο„Έ 1 π‘₯ π‘₯ 3 βˆ’ 2 4 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q12:

Determine whether the integral ο„Έ 𝑀 𝑀 βˆ’ 2 𝑀 5 0 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q13:

Determine whether the integral ο„Έ 1 π‘₯ βˆ’ π‘₯ βˆ’ 2 π‘₯ 4 0 2 d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q14:

The integral ο„Έ 1 √ π‘₯ βˆ’ 1 π‘₯ 9 0 3 d is convergent. What does it converge to?

  • A βˆ’ 1 2
  • B2
  • C βˆ’ 1 5 2
  • D 9 2
  • E βˆ’ 1

Q15:

Consider the integral ο„Έ π‘₯ π‘₯ π‘₯    l n d .

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

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

Evaluate the integral for those values of 𝑝 .

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

Q16:

Consider the integral ο„Έ 1 π‘₯ π‘₯    d .

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

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

Evaluate the integral for those values of 𝑝 .

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

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