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### Review of Trigonometric Integrals

Course: MATH 201, Spring 2011
School: Qatar University
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Word Count: 945

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Integrals I. Review Trigonometric cosnx dx Case 1: n is an odd natural number 1. cos x dx = sin x + c 2. cos 3 x dx = cos 2 x cos x dx = (1 sin 2 x) cos x dx = (cos x sin 2 x cos x) dx sin 3 x = sin x +c 3 1* . sin 20 x cos x dx sin 21 x = +c 21 2 *. sin 20 3 x cos x dx = sin 20 x cos 2 x cos x dx = sin 20 x (1 sin 2 x) cos x dx = (sin 20 x sin 22 x ) cos x dx sin 21 x sin 23 x = +c...

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Integrals I. Review Trigonometric cosnx dx Case 1: n is an odd natural number 1. cos x dx = sin x + c 2. cos 3 x dx = cos 2 x cos x dx = (1 sin 2 x) cos x dx = (cos x sin 2 x cos x) dx sin 3 x = sin x +c 3 1* . sin 20 x cos x dx sin 21 x = +c 21 2 *. sin 20 3 x cos x dx = sin 20 x cos 2 x cos x dx = sin 20 x (1 sin 2 x) cos x dx = (sin 20 x sin 22 x ) cos x dx sin 21 x sin 23 x = +c 21 23 cos 3 x 2 * *. 3 = sin 2 3 = sin 2 3 = (sin 1 3 dx sin 2 x x cos 2 x cos x dx x (1 sin 2 x) cos x dx 2 3 4 3 x sin x ) cos x dx 7 3 sin x sin x = +c 1 7 3 3 3. cos 5 x dx = cos x cos x dx 4 2 = (1 sin x) cos x dx 2 = (1 2 sin x + sin x ) cos x dx 2 4 3 5 sin x sin x = sin x 2 + +c 3 5 4. cos 7 x dx = cos x cos x dx 6 3 = (1 sin x) cos x dx 2 = (1 3 sin x + 3 sin x sin x ) cos x dx 2 4 3 6 5 7 sin x 3 sin x sin x = sin x 3 + +c 3 5 7 5 7 3 sin x sin x 3 = sin x sin x + +c 5 7 I. cos x dx n Case 2: n is an even natural number 1. cos 2 x dx 1 + cos 2 x = dx 2 1 = dx + cos 2 x dx 2 1 11 = x + cos 2 x 2dx 2 22 1 1 = x + sin 2 x + c 2 4 [ ] 2. cos 4 x dx 2 1 + cos 2 x I = dx 2 1 = 1 + 2 cos 2 x + cos 2 2 x dx 4 1 2 = dx + cos 2 x 2dx + cos 2 x dx 4 1 1 1 2 = x+ sin 2 x + cos 2 x dx 4 4 4 [ [ ] ] I 1 = cos 2 2 x dx 1 + cos 4 x dx 2 1 = dx + cos 4 x dx 2 1 11 = x + cos 4 x 4dx 2 24 1 11 = x + sin 4 x + c 2 24 Thus, = [ ] 1 1 11 1 I = x+ sin 2 x + x + sin 4 x + c 4 4 42 8 3 1 1 = x+ sin 2 x + sin 4 x + c 8 4 32 3. cos 2 x sin 2 x dx 1 + cos 2 x 1 cos 2 x = dx 2 2 1 = 1 cos 2 2 x dx 4 1 1 1 + cos 4 x = x dx 4 4 2 1 1 1 = x x + cos 4 x 4dx 2 8 4 3 1 = x sin 4 x + c 8 32 ( ) II. tannx dx Case 1: n is an odd natural number 1. tan x dx sin x = dx cos x sin x = dx cos x = ln cos x + c 2. tan 3 x dx = tan x tan x dx 2 = tan x (sec x 1) dx 2 = (tan x sec 2 x tan x) dx 1 = tan 2 x + ln cos x + c 2 3. tan 5 x dx = 3 tan x tan 2 x dx = tan 3 x (sec 2 x 1) dx = (tan 3 x sec 2 x tan 3 x) dx = tan 3 x sec 2 x dx tan 3 x dx 1 1 4 = tan x tan 2 x ln cos x + c 4 2 secmx tannx dx where n is odd 1. sec x tan x dx = sec x + c 2. sec x tan x dx 20 = sec x sec x tan x dx 19 20 sec x = +c 20 3. sec 20 3 x tan x dx = sec x (sec x 1) sec x tan x dx 19 2 = (sec x sec x) sec x tan x dx 21 22 19 20 sec x sec x = +c 22 20 4. 2 3 5 sec x tan x dx 2 3 = sec x (sec 2 x 1) 2 tan x dx = sec = sec 1 3 1 3 x (sec 2 x 1) 2 tan x sec x dx x (sec 4 x 2 sec 2 x + 1) tan x sec x dx 11 3 5 3 = (sec x 2 sec x + sec 14 3 8 3 2 3 1 3 x) tan x sec x dx sec x sec x sec x = 2 + +c 11 8 2 3 3 3 II. tan x dx n Case 2: n is an even natural number 1. tan x dx 2 = (sec x 1 ) dx 2 = tan x x + c 2. tan 4 x dx = tan 2 x (sec 2 x 1 ) dx = (tan 2 x sec 2 x tan 2 x ) dx tan 3 x = tan 2 x d x 3 tan 3 x = tan x + x + c 3 3. tan 6 x dx = tan x (sec x 1 ) dx 4 2 = (tan 4 x sec 2 x tan 4 x ) dx 5 tan x = tan 4 x d x 5 5 3 tan x tan x = + tan x x + c 5 3 secnx dx where n is even 1. sec x dx 2 = tan x + c 2. sec x dx 4 = (1 + tan x) sec x dx 2 2 = sec x dx + tan x sec x dx 2 2 3 tan x = tan x + +c 3 2 3. sec x dx 6 = (1 + tan x) sec x dx 2 2 2 = (1 + 2 tan x + tan x) sec x dx 2 4 3 2 5 tan x tan x = tan x + 2 + +c 3 5 4. sec 8 x dx = (1 + tan x) sec x dx 2 3 2 = (1 + 3 tan x + 3 tan x + tan x) sec x dx 2 4 3 6 5 2 7 tan x tan x tan x = tan x + 3 + 3 + +c 3 5 7 5 7 tan x tan x 3 = tan x + tan x + 3 + +c 5 7 tanmx secnx dx where n is even 1. 3 tan 5 x sec 4 x dx 5 3 = tan x (1 + tan 2 x) sec 2 x dx 5 3 = (tan x + tan 8 3 14 3 11 3 x) sec 2 x dx tan x tan x = + +c 8 14 3 3 secnx where n is odd 1. sec x dx sec x(tan x + sec x) = dx tan x + sec x 2 sec x tan x + sec x = dx tan x + sec x = ln tan x + sec x + c 2. 3. sec sec 3 x dx 5 x dx etc.. We use int egration by parts
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