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 𝑒π‘₯βˆžοŠ¦οŠ±ο—οŽ’d an improper integral?

  • Ano
  • Byes

Q2:

Consider the integral ο„Έ1π‘₯(π‘₯)π‘₯∞lnd.

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

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

Evaluate the integral for those values of 𝑝.

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

Q3:

The integral ο„Έ1π‘₯π‘₯π‘₯∞lnd is convergent. What does it converge to?

Q4:

The integral ο„Έπ‘₯π‘₯π‘₯∞lnd is convergent. What does it converge to?

Q5:

Determine whether the integral ο„Έπ‘’πœƒπœƒβˆžοŠ¦οΌcossind is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q6:

Determine whether the integral ο„Έπ‘₯√1+π‘₯π‘₯∞d is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q7:

The integral ο„Έπ‘’π‘βˆžοŠ¨οŠ±οŠ«οŒd is convergent. What does it converge to?

  • A 𝑒 5   
  • B 5 𝑒   
  • C βˆ’ 5 𝑒  
  • D βˆ’ 𝑒 5   
  • E βˆ’ 𝑒 5  

Q8:

The integral ο„Έπ‘’π‘¦βˆžοŠ¦οŠ±βˆšο˜d is convergent. What does it converge to?

Q9:

Determine whether the integral ο„Έπ‘₯π‘₯π‘₯∞lnd is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q10:

Consider the integral ο„Έ1(2π‘₯+1)π‘₯∞d.

Determine whether the integral is convergent or divergent.

  • ADivergent
  • BConvergent

Determine the value of integration.

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

Q11:

The integral ο„Έ1π‘₯+π‘₯π‘₯∞d is convergent. What does it converge to?

  • A βˆ’ 2 l n
  • B0
  • C 1 2 l n
  • D βˆ’ 1 2 l n
  • E l n 2

Q12:

The integral ο„Έ2π‘ŸοŠ¦οŠ±βˆžοŽd 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„Έ13βˆ’4π‘₯π‘₯∞d is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q14:

The integral 𝑣𝑣+2π‘£βˆ’3∞d is convergent. What does it converge to?

  • A 4 5 l n
  • B βˆ’ 5 4 l n
  • C0
  • D l n 5 4
  • E βˆ’ 4 5 l n

Q15:

Determine whether the integral ο„Έο€Ήπ‘¦βˆ’3π‘¦ο…π‘¦βˆžοŠ±βˆžοŠ©οŠ¨d is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q16:

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

  • A πœ‹ 4
  • B 3 πœ‹ 4
  • C 3 πœ‹ 2
  • D0
  • E πœ‹ 2

Q17:

The integral ο„Έπ‘₯𝑒π‘₯βˆžοŠ±βˆžοŠ±ο—οŽ‘d is convergent. What does it converge to?

Q18:

The integral 𝑧𝑧+4π‘§οŠ¦οŠ±βˆžοŠͺd is convergent. What does it converge to?

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

Q19:

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

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

Q20:

The integral ο„Έπ‘§π‘’π‘§οŠ¦οŠ±βˆžοŠ¨ο™d is convergent. What does it converge to?

  • A2
  • B 1 4
  • C βˆ’ 1 4
  • D βˆ’ 2
  • E 3 4

Q21:

Determine whether the integral ο„Έ1√1+π‘₯π‘₯∞d is convergent or divergent.

  • Adivergent
  • Bconvergent

Q22:

The integral ο„Έ1(π‘₯βˆ’2)π‘₯∞d is convergent. What does it converge to?

Q23:

The integral ο„Έπ‘¦π‘’π‘¦βˆžοŠ¨οŠ±οŠ©ο˜d is convergent. What does it converge to?

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

Q24:

Determine whether the integral ο„Έπ›Όπ›ΌβˆžοŠ¦οŠ¨sind is convergent or divergent.

  • Aconvergent
  • Bdivergent

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