Reduction Formulas (Secant and Cosecant)

PROOF

(ii) This can be done by integration by parts, but it is easier to use Theorem 2. Let n = 2 - m. For m ≠ 2, n ≠ 0 and Theorem 2 gives

whence

For m = 2 the formula is already known,

These reduction formulas can be used to integrate any even power of sec x or csc x, and to get the integral of any odd power of sec x or csc x in terms of ∫ sec x or ∫ csc x. We shall find ∫ sec x and ∫ csc x in the next chapter.