MTH 252 Integral Calculus Chapter 8 – Principles of Integral Evaluation Section 8.3 – Trigonometric Integrals Copyright © 2005 by Ron Wallace, all rights reserved.

Odd Powers of SIN & COS n > 2 is a positive odd integer n - 1 is a positive even integer Multiply out the polynomial, integrate, and substitute back.

Odd Powers of SIN & COS n > 2 is a positive odd integer n - 1 is a positive even integer Multiply out the polynomial, integrate, and substitute back.

Odd Powers of SIN & COS m is a positive integer n is a positive odd integer n - 1 is a positive even integer Multiply out the polynomial, integrate, and substitute back.

Odd Powers of SIN & COS m is a positive odd integer n is a positive integer m - 1 is a positive even integer Multiply out the polynomial, integrate, and substitute back.

Powers of SIN & COS NOTE: m & n are non-negative integers. If n is odd, put one of the cosines w/ dx, change the remaining cosines to sines, and let u = sin x. If m is odd, put one of the sines w/ dx, change the remaining sines to cosines, and let u = cos x. All other cases … use some other method.

Powers of TAN & SEC For what values of m & n can the same approach be used? [NOTE: m & n are non-negative integers.] If m is odd and n > 0, put one of the tangents and one of the secants w/ dx, change the remaining tangents to secants, and let u = sec x. If n is even and n > 0, put two of the secants w/ dx, change the remaining secants to tangents, and let u = tan x. All other cases … use some other method.

Even Powers of SIN & COS m and n are BOTH non-negative even integers. Remember the half-angle formulas:

Even Powers of SIN & COS Use the trigonometric identities … m and n are BOTH non-negative even integers. Use the trigonometric identities … Multiply everything out. Integrate each term, one at a time.

Powers of SEC Memorize this one!

Use Integration by Parts Powers of SEC Use Integration by Parts n > 2 and a positive integer

Powers of SEC Use Integration by Parts n > 2 and a positive integer This kind of identity is called a “Reduction Formula.” No! You do NOT need to memorize this one. Just use it.

Powers of TAN Memorize this one too!

n > 1 and a positive integer Powers of TAN n > 1 and a positive integer If n is even, change to secants using … If n is odd, Put sec x tan x w/ dx as follows … Convert all tangents (except the one w/ dx) to secants. Use the substitution, u = sec x

FYI: More Reduction Formulas

An equivalent form to the other solution. One more note … An equivalent form to the other solution.