Use the substitution u = sec x to find the exact value of ∫(from 0 to π/3) (tan x)(sec^3 x) dx.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1: States or implies du/dx = sec x tan x or dx/du = 1/(sec x tan x).
2. M1: Substitutes u = sec x and du/dx into the integral to express it in terms of u.
3. A1: Correctly transforms the integral to ∫ u^2 du.
4. B1: Correctly changes the limits of integration. Lower limit: u = sec(0) = 1. Upper limit: u = sec(π/3) = 2.
5. M1: Integrates u^2 with respect to u to get (1/3)u^3.
6. A1: Evaluates the definite integral using the correct limits to get the exact value (7/3).
7. Model Answer:
8. Let u = sec x
9. du/dx = sec x tan x
10. dx = du / (sec x tan x)
11. Substitute into the integral:
12. ∫ (tan x)(sec^3 x) dx = ∫ (tan x)(u^3) [du / (u tan x)]
13. = ∫ u^2 du
14. Change limits:
15. When x = 0, u = sec(0) = 1
16. When x = π/3, u = sec(π/3) = 1/cos(π/3) = 1/(1/2) = 2
17. So, ∫(from 1 to 2) u^2 du
18. = [u^3 / 3] (from 1 to 2)
19. = (2^3 / 3) - (1^3 / 3)
20. = 8/3 - 1/3 = 7/3