Trigonometric Integrals and Substitutions This section consists of two parts: Part I: Trigonometric Integrals. We will distinguish three main cases: Case A: Integrals of type where and are nonnegative integers. METHOD OF INTEGRATION: (i)  If is odd, then (ii)  If is odd, then (iii)  If both and are even, then use the identities or (and) sometimes EXAMPLE 1: Find Solution: We have EXAMPLE 2: Find Solution: We have EXAMPLE 3: Find Solution: We have EXAMPLE 4: Find Solution: We have EXAMPLE 5: Find Solution: We have EXAMPLE 6: Find Solution: We have Case B: Integrals of type where and are nonnegative integers. METHOD OF INTEGRATION: (i)  If is odd, then (ii)  If is even, then (iii)  In other cases the guidelines are not as clear-cut. EXAMPLE 7: Find Solution 1: We have Solution 2: We have EXAMPLE 8: Find (a)                          (b) Solution 1: (a) We have In short, This can be rewritten as (see Appendix II). (b) We have which is by (a). Substituting for we get This can be rewritten as (see Appendix II). Solution 2: (a) We have This can be rewritten as (see Appendix II). (b) We have This can be rewritten as (see Appendix II). Solution 3: (a) We have (b) We have EXAMPLE 9: Find Solution: We have EXAMPLE 10: Find Solution 1: We have Solution 2: We have To evaluate we either or Therefore Other possible answers are EXAMPLE 11: Find Solution: We have It follows that so Case C: Integrals of type METHOD OF INTEGRATION: Use the identities EXAMPLE 12: Find Solution 1: We have Solution 2: We have therefore hence Part II: Trigonometric Substitutions. Here we deal with integrals that involve METHOD OF INTEGRATION: (i)  If , then (ii)  If , then (iii)  If , then or EXAMPLE 13: Find Solution 1: We have Note that Therefore Solution 2: We have Therefore so EXAMPLE 14: Find Solution: We have EXAMPLE 15: Find Solution: We have EXAMPLE 16: Find Solution 1: We have Note that                                                                                     Therefore Solution 2: We have EXAMPLE 17: Find Solution 1: We have Note that                              Therefore Solution 2: We first note that We have Therefore Solution 3 (version 1): Note that can be rewritten as with It is known that if then Therefore we have Solution 3 (version 2): We have EXAMPLE 18: Find Solution: We have Since we get We also note that since we have if then              and             if then Therefore EXAMPLE 19: Find Solution 1: We have Solution 2: We have Note that                             Therefore EXAMPLE 20: Find Solution: We have Note that and therefore Appendix I EXAMPLE 21: Find Solution 1: We have therefore so Solution 2: We have EXAMPLE 22: Find Solution: We have therefore so In particular, if we have Alternatively, Similarly, if we have Alternatively, Appendix II