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)