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 sin x dx n 2 is a positive odd integer sin n 1 x sin x dx sin x 2 n 1 2 sin x dx 1 cos x 2 1 u n 1 2 n 1 2 2 n 1 is a positive even integer sin x dx du u cos x du sin x dx Multiply out the polynomial integrate and substitute back Odd Powers of SIN COS n cos x dx n 2 is a positive odd integer cosn 1 x cos x dx cos x 2 n 1 2 cos x dx 1 sin x 2 1 u n 1 2 2 n 1 is a positive even integer n 1 2 du cos x dx u sin x du cos x dx Multiply out the polynomial integrate and substitute back Odd Powers of SIN COS m is a positive integer n is a positive odd integer m n sin n cos x dx n 1 is a positive sin m x cosn 1 x cos x dx sin x cos x m 2 n 1 2 sin x 1 sin x m u 1 u m 2 n 1 2 2 du even integer cos x dx n 1 2 cos x dx u sin x du cos x dx 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 n sin n cos x dx m 1 is a positive sin m 1 x cosn x sin x dx sin x 2 m 1 2 1 cos x 2 1 u n cos x sin x dx m 1 2 m 1 2 2 even integer n n cos x sin x dx u du u cos x du sin x dx Multiply out the polynomial integrate and substitute back Powers of SIN COS m n sin n cos x dx 1 sin 2 x cos2 x 1 cos2 x sin 2 x NOTE m n are non negative integers d d Since sin x cos x cos x sin x dx dx 1 If n is odd put one of the cosines w dx change the remaining cosines to sines and let u sin x 2 If m is odd put one of the sines w dx change the remaining sines to cosines and let u cos x 3 All other cases use some other method Powers of TAN SEC tan m n n sec x dx 1 tan 2 x sec 2 x sec 2 x 1 tan 2 x For what values of m n can the same approach be used NOTE m n are non negative integers d d 2 Hint tan x sec x sec x sec x tan x dx dx 1 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 2 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 3 All other cases use some other method Even Powers of SIN COS m n sin n cos x dx m and n are BOTH non negative even integers Remember the half angle formulas cos q 1 cos q 2 2 sin q 1 cos q 2 2 Let q 2x 1 cos 2 x cos x 2 sin x 1 cos 2 x 2 Square both sides 1 cos 2 x cos2 x 2 1 cos 2 x sin 2 x 2 Even Powers of SIN COS m n sin n cos x dx m and n are BOTH non negative even integers 1 Use the trigonometric identities cos2 x 1 1 cos 2 x 2 sin 2 x 1 1 cos 2 x 2 2 Multiply everything out 3 Integrate each term one at a time Powers of SEC sec x dx sec x tan x sec x dx sec x tan x sec 2 x sec x tan x dx sec x tan x Let u sec x tan x du sec x tan x sec 2 x 1 du ln u c u ln sec x tan x c Memorize this one Powers of SEC sec n sec x dx ln sec x tan x c sec x dx tan x c 2 Use Integration by Parts x dx n 2 and a positive integer u sec n 2 x dv sec 2 x dx du n 2 sec n 2 x tan x dx v tan x n n 2 n 2 2 sec x dx sec x tan x n 2 sec x tan x dx n n 2 n 2 2 sec x dx sec x tan x n 2 sec x sec x 1 dx n n 2 n n 2 sec x dx sec x tan x n 2 sec x dx n 2 sec x dx n 1 sec n x dx sec n 2 x tan x n 2 sec n 2 x dx Powers of SEC sec n sec x dx ln sec x tan x c sec x dx tan x c 2 Use Integration by Parts x dx n 2 and a positive integer u sec n 2 x dv sec 2 x dx du n 2 sec n 2 x tan x dx v tan x 1 n 2 n 2 n 2 sec x dx sec x tan x sec x dx n 1 n 1 n This kind of identity is called a Reduction Formula n 1 sec n x dx sec n 2 x tan x n 2 sec n 2 x dx No You do NOT need to memorize this one Just use it Powers of TAN tan x dx sin x dx cos x Let u cos x du sin x dx 1 du ln u c ln cos x c u ln sec x c Memorize this one too tan x dx ln sec x c Powers of TAN n tan x dx n 1 and a positive integer 1 If n is even change to secants using tan 2 x sec 2 x 1 2 If n is odd a Put sec x tan x w dx as follows n 1 tan x n tan x dx sec x sec x tan x dx b Convert all tangents except the one w c Use the substitution u sec x dx to secants FYI More Reduction Formulas 1 n 1 n 1 n 2 sin x dx sin x cos x sin x dx n n n 1 n 1 n 1 n 2 cos x dx cos x sin x cos x dx n n n 1 n 1 n 2 tan x dx tan x tan x dx n 1 n 1 n 2 n 2 n 2 sec x dx sec x …