5.1 Reduction formulae Quiz: Pearson Edexcel Further Maths, Unit 5

The 20 questions

1. What is a reduction formula in integral calculus?

   • A rule for decomposing a rational function into partial fractions
   • A formula for the sum of a finite arithmetic series
   • A formula expressing I_n in terms of a simpler integral such as I_(n-1) or I_(n-2)
   • A formula giving the derivative of an inverse trigonometric function

2. Which integration technique is usually used to derive a reduction formula for I_n = integral of x^n e^x dx?

   • Differentiation under the integral sign
   • Partial fractions only
   • Integration by parts
   • Substitution u = x^2 only

3. What is the value of the integral from 0 to pi/2 of sin^0 x dx?

   • 1
   • pi
   • pi/2
   • 0

4. What is the value of the integral from 0 to pi/2 of sin x dx?

   • 0
   • -1
   • 1
   • pi/2

5. What is the value of the integral from 0 to pi/2 of sin^2 x dx?

   • pi/2
   • 1/2
   • pi/4
   • pi

6. For I_n = integral from 0 to pi/2 of sin^n x dx, which reduction formula holds?

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

7. Using I_n = integral from 0 to pi/2 of sin^n x dx, what is I_4?

   • 3/4
   • 3pi/8
   • pi/4
   • 3pi/16

8. For I_n = integral from 0 to 1 of x^n e^x dx, what is I_2?

   • 2 - e
   • e - 2
   • e + 2
   • e - 1

9. For I_n = integral from 0 to 1 of x^n e^x dx, what is I_3?

   • 6 - 2e
   • 6 - e
   • 2e - 6
   • e - 6

10. For the indefinite integral I_n = integral of x^n e^x dx, which reduction formula holds?

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

11. For I_n = integral of tan^n x dx, what is I_2 up to a constant?

    • sec^2 x
    • tan x - x
    • x - tan x
    • tan^2 x / 2

12. What is the value of the integral from 0 to pi/2 of cos^5 x dx?

    • pi/5
    • 8/15
    • 15/8
    • 4/5

13. What is the value of the integral from 0 to pi/2 of sin^3 x dx?

    • 3/2
    • pi/3
    • 2/3
    • 1/3

14. What is the value of the integral from 0 to pi/2 of sin^6 x dx?

    • 5pi/16
    • 5pi/32
    • pi/32
    • 15pi/32

15. For I_n = integral from 0 to pi/2 of sin^n x dx, what is I_7?

    • 6/7
    • 16/35
    • 32/35
    • 8/35

16. For I_n = integral from 1 to e of (ln x)^n dx, which reduction formula holds?

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

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

    • 4
    • 24
    • 120
    • 16

18. For I_n = integral from 0 to 1 of x^n e^x dx, what is I_4?

    • 9e - 24
    • 24 - 9e
    • 9e - 6
    • e - 24

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

    • 4/15
    • 8/15
    • 1/5
    • 2/3

20. As n increases, how does the integral I_n = integral from 0 to pi/2 of sin^n x dx behave?

    • It stays constant at pi/2
    • It decreases towards 0
    • It alternates in sign
    • It increases without bound

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