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


EXAMPLE 2: Find


EXAMPLE 3: Find


EXAMPLE 4: Find


EXAMPLE 5: Find


EXAMPLE 6: Find


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


EXAMPLE 8: Find

(a)                          (b)


EXAMPLE 9: Find


EXAMPLE 10: Find


EXAMPLE 11: Find


Case C: Integrals of type



METHOD OF INTEGRATION: Use the identities



EXAMPLE 12: Find


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


EXAMPLE 14: Find


EXAMPLE 15: Find


EXAMPLE 16: Find


EXAMPLE 17: Find


EXAMPLE 18: Find


EXAMPLE 19: Find


EXAMPLE 20: Find



Appendix I

EXAMPLE 21: Find


EXAMPLE 22: Find




Appendix II