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Theorem 1

7.5 INTEGRALS OF POWERS OF TRIGONOMETRIC FUNCTIONS

It is often possible to transform an integral into one of the forms

∫ sinn u du,           ∫ cosn u du,           ∫ tann u du, etc.

These integrals can be evaluated by means of reduction formulas, which express the integral of the nth power of a trigonometric function in terms of the (n - 2)nd power. The easiest reduction formulas to prove are those for the tangent and cotangent, so we shall give them first.

THEOREM 1

Let n ≠ 1.

Then

(i)

07_trigonometric_functions-279.gif

(ii)

07_trigonometric_functions-280.gif

PROOF

We recall that

tan2 x = sec2 x - 1, d(tan x) = sec2 x dx.

Then

07_trigonometric_functions-281.gif

These reduction formulas are true for any rational number n ≠ 1. They are most useful, however, when n is a positive integer.
Home Trigonometric Functions Integrals of Powers of Trigonometric Functions Theorem 1

Last Update: 2006-11-05