E8-E9 Reduction formulae and limits Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Which reduction formula holds for I_n = integral from 0 to pi/2 of sin^n(x) dx, n >= 2?

   • I_n = (n/(n - 1)) I_(n - 2)
   • I_n = ((n - 1)/n) I_(n - 1)
   • I_n = ((n + 1)/n) I_(n - 2)
   • I_n = ((n - 1)/n) I_(n - 2)

2. Evaluate the integral from 0 to pi/2 of sin^2(x) dx.

   • pi/4
   • pi/8
   • 1/2
   • pi/2

3. Evaluate the integral from 0 to pi/2 of sin^3(x) dx.

   • pi/3
   • 3/4
   • 2/3
   • 1/3

4. Evaluate the integral from 0 to pi/2 of sin^4(x) dx.

   • 3/16
   • 3 pi/16
   • pi/8
   • 3 pi/8

5. For I_n = integral of x^n e^x dx, which reduction formula is correct?

   • I_n = n x^(n - 1) e^x - I_(n - 1)
   • I_n = x^n e^x + n I_(n - 1)
   • I_n = x^n e^x - n I_(n - 1)
   • I_n = x^n e^x - I_(n - 1)/n

6. For I_n = integral of tan^n(x) dx, which reduction formula is correct?

   • I_n = tan^(n + 1)(x)/(n + 1) - I_(n + 2)
   • I_n = tan^(n - 1)(x)/(n - 1) + I_(n - 2)
   • I_n = tan^n(x)/n - I_(n - 2)
   • I_n = tan^(n - 1)(x)/(n - 1) - I_(n - 2)

7. Evaluate the limit of x^2 e^(-x) as x tends to infinity.

   • 1
   • 2
   • 0
   • infinity

8. Evaluate the improper integral of x e^(-x) from 0 to infinity.

   • 1
   • 2
   • 0
   • -1

9. Evaluate the improper integral of x^2 e^(-x) from 0 to infinity.

   • 3
   • 2
   • 1
   • 4

10. Evaluate the improper integral of e^(-2x) from 0 to infinity.

    • 1
    • 1/2
    • 2
    • 1/4

11. Evaluate the improper integral of ln(x) from 0 to 1.

    • 1/2
    • 0
    • -1
    • 1

12. Evaluate the improper integral of x ln(x) from 0 to 1.

    • -1/2
    • 0
    • -1/4
    • 1/4

13. Consider the improper integral of 1/x^2 from 1 to infinity. What is its value?

    • 1
    • 2
    • 1/2
    • The integral diverges

14. Does the improper integral of 1/x from 1 to infinity converge?

    • No, it diverges to infinity
    • Yes, it equals 1
    • Yes, it equals ln(2)
    • Yes, it equals 0

15. What is the limit of x ln(x) as x tends to 0 from the positive side?

    • -infinity
    • -1
    • 0
    • 1

16. Using integration by parts, the integral of x^n e^x is reduced. Evaluate the integral from 0 to 1 of x^2 e^x dx.

    • e
    • e - 2
    • e + 2
    • 2 - e

17. Evaluate the improper integral of x^3 e^(-x) from 0 to infinity.

    • 9
    • 2
    • 6
    • 3

18. Evaluate the integral from 0 to pi/2 of sin^5(x) dx.

    • 1/5
    • 4/15
    • 4/5
    • 8/15

19. Evaluate the integral from 0 to pi/2 of cos^6(x) dx.

    • 5 pi/16
    • pi/32
    • 3 pi/32
    • 5 pi/32

20. Evaluate the improper integral of 1/x^3 from 1 to infinity.

    • 3
    • 1/3
    • 1
    • 1/2

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