IV. INTEGRATION OF TRIGONOMETRIC FUNCTIONS

Integrating the following with respect to x

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. Find the following integrals:

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. (a) Show that , and hence find (b) Show that , and hence find (c) Prove that , and hence find (d) Prove by differentiation that = or . Show that each of the primitive functions differ only by a constant.

30. Find the following integrals by using the substitution method:

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) 31. Determine the value of the following integrals:

(a) (b) (c) (d) (e) (f) (g) (h) 32. (a) Find hence find .

(b) Show that hence find (c) Show that hence find (d) Differentiate and ; make use of these results to find .

(e) Show that and hence find .

(f) Show that , and hence find (g) Differentitate and hence find .

33. (a) For what values of a does (i) and

(ii) (b) Find (i) (ii) (c) Prove that (i) (ii) (iii) , hence find the following integrals: , , .

(d) Show that (i) (ii) (iii) (iv) 