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.

  • A𝑝>1
  • B𝑝≀1
  • C𝑝<1
  • D𝑝β‰₯1
  • E𝑝=1

Evaluate the integral for those values of 𝑝.

  • Aπ‘βˆ’1
  • B1𝑝
  • C1𝑝+1
  • D𝑝+1
  • E1π‘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?

  • A𝑒5
  • B5π‘’οŠ±οŠ§οŠ¦
  • Cβˆ’5π‘’οŠ±οŠ«
  • Dβˆ’π‘’5
  • Eβˆ’π‘’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.

  • A136
  • Bβˆ’136
  • C518
  • D14
  • Eβˆ’29

Q11:

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

  • Aβˆ’2ln
  • B0
  • C12ln
  • Dβˆ’12ln
  • Eln2

Q12:

The integral ο„Έ2π‘ŸοŠ¦οŠ±βˆžοŽd is convergent. What does it converge to?

  • A22ln
  • Bβˆ’12ln
  • Cβˆ’2ln
  • D12ln
  • E22ln

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?

  • A45ln
  • Bβˆ’54ln
  • C0
  • Dln54
  • Eβˆ’45ln

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?

  • Aπœ‹4
  • B3πœ‹4
  • C3πœ‹2
  • D0
  • Eπ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?

  • Aπœ‹2
  • Bβˆ’πœ‹2
  • Cβˆ’πœ‹8
  • Dβˆ’πœ‹16
  • Eπœ‹8

Q19:

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

  • A1𝑒
  • B1π‘’βˆ’1
  • C1βˆ’π‘’
  • D1βˆ’1𝑒
  • E1+1𝑒

Q20:

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

  • A2
  • B14
  • Cβˆ’14
  • Dβˆ’2
  • E34

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?

  • A63π‘’οŠ±οŠ¬
  • B7𝑒9
  • Cβˆ’63π‘’οŠ±οŠ©
  • D7𝑒9
  • Eβˆ’7𝑒9

Q24:

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

  • Aconvergent
  • Bdivergent

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