5.2 Improper integrals Quiz: Pearson Edexcel Further Maths, Unit 5

The 20 questions

1. What makes an integral improper?

   • An integrand that is continuous on a finite closed interval
   • A result that is negative
   • An integrand that is a polynomial of degree at least two
   • An infinite limit of integration, or an integrand that is unbounded within the range

2. For which values of p does the integral from 1 to infinity of 1/x^p dx converge?

   • p > 1
   • p = 1 only
   • p < 1
   • p >= 0

3. What is the behaviour of the integral from 1 to infinity of 1/x dx?

   • It converges to 1
   • It converges to 0
   • It diverges
   • It converges to ln 2

4. What is the value of the integral from 0 to infinity of e^(-x) dx?

   • 1
   • e
   • 0
   • infinity

5. What is the value of the integral from 1 to infinity of 1/x^2 dx?

   • 1/2
   • 2
   • infinity
   • 1

6. What is the value of the integral from 0 to 1 of 1/sqrt(x) dx?

   • 1
   • infinity
   • 2
   • 1/2

7. What is the value of the integral from 0 to 1 of ln x dx?

   • 1
   • -1
   • -2
   • 0

8. What is the value of the integral from 0 to infinity of 1/(1 + x^2) dx?

   • 1
   • pi
   • pi/2
   • pi/4

9. What is the behaviour of the integral from 0 to 1 of 1/x^2 dx?

   • It converges to 2
   • It diverges
   • It converges to 1
   • It converges to -1

10. What is the value of the integral from 2 to infinity of 1/x^3 dx?

    • 1/2
    • 1/16
    • 1/8
    • 1/4

11. What is the value of the integral from 0 to infinity of x e^(-x) dx?

    • -1
    • 1
    • 0
    • 2

12. What is the value of the integral from 0 to infinity of e^(-2x) dx?

    • 1/2
    • 2
    • -1/2
    • 1

13. What is the value of the integral from 0 to 1 of x^(-1/3) dx?

    • 3/2
    • 2/3
    • 1/3
    • infinity

14. What is the value of the integral from 0 to 4 of 1/sqrt(4 - x) dx?

    • infinity
    • 4
    • 8
    • 2

15. Which of the following integrals converges?

    • The integral from 0 to infinity of 1 dx
    • The integral from 1 to infinity of 1/x dx
    • The integral from 1 to infinity of x^(-3/2) dx
    • The integral from 1 to infinity of x^(-1/2) dx

16. What is the value of the integral from 0 to infinity of x^2 e^(-x) dx?

    • 6
    • 1
    • 4
    • 2

17. What is the value of the integral from 0 to infinity of x/(1 + x^2)^2 dx?

    • 1/4
    • 1/2
    • 1
    • It diverges

18. What is the value of the integral from 0 to 1 of (ln x)^2 dx?

    • -2
    • -1
    • 2
    • 1

19. Is the integral from 0 to infinity of sin x dx convergent?

    • Yes, it converges to 2
    • Yes, it converges to 0 by symmetry
    • No, its partial integrals oscillate and have no limit
    • Yes, it converges to 1

20. If the integral from 1 to infinity of k/x^3 dx equals 4, what is k?

    • 16
    • 8
    • 2
    • 4

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