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Trigonometric Integrals I. Integrating Powers of the Sine and Cosine Functions A. Useful trigonometric identities 1. sin 2 x cos 2 x 1 2. sin 2x 2 sin x cos x 3. cos 2 x cos 2 x sin 2 x 2 cos 2 x 1 1 2 sin 2 x 1 cos 2 x 2 1 cos 2 x cos 2 x 2 1 sin x cos y [sin( x y ) sin( x y )] 2 1 sin x sin y [cos( x y ) cos( x y )] 2 1 cos x cos y [cos( x y ) cos( x y )] 2 4. sin 2 x 5. 6. 7. 8. B. Reduction formulas 1. 1 n 1 sin n x dx sin n1 x cos x sin n2 x dx n n 2. cos n x dx 1 n 1 cos n1 x sin x cos n2 x dx n n C. Examples 1. Find sin 2 x dx . sin x (sin x dx) . Let u sin x and dv sin x dx du cos x dx and v sin x dx Method 1(Integration by parts): 1 sin 2 x dx cos x . Thus, sin 2 x dx (sin x)( cos x) cos 2 x dx sin x cos x (1 sin x) dx sin x cos x 1dx sin x dx sin x cos x x sin x dx 2 sin x dx sin x cos x x sin x dx 2 2 2 2 2 1 1 sin x cos x x C . 2 2 Method 2(Trig identity): sin 2 x dx Method 3(Reduction formula): 1 1 1 (1 cos 2 x) dx x sin 2 x C . 2 2 4 1 1 sin 2 x dx sin x cos x 1dx 2 2 1 1 sin x cos x x C . 2 2 2. Find cos x dx . 3 Use the reduction formula: 1 2 cos 3 x dx cos 2 x sin x cos x dx 3 3 1 2 1 2 cos 2 x sin x sin x C sin x(1 sin 2 x) sin x C 3 3 3 3 1 sin x sin 3 x C . 3 3. Find sin 3 x cos 2 x dx . sin x cos x dx sin x sin x cos x dx (1 cos x) cos x sin xdx (cos x cos x)(sin x dx) . Let u cos x du sin x dx . Thus, 3 2 2 2 2 2 2 4 1 1 1 (cos 2 x cos 4 x)(sin x dx) (u 2 u 4 ) du u 3 u 5 C 3 5 1 1 cos 3 x cos 5 x C . 3 5 2 4. Find sin x cos x dx . 2 2 1 1 cos 2 x 1 cos 2 x (1 cos 2 2 x) dx dx 2 2 4 1 1 1 cos 4 x 1 1 sin 2 2 x dx 1 dx cos 4 x dx dx 4 4 2 8 8 1 1 x sin 4 x C . 8 32 sin 2 x cos 2 x dx 5. Find sin 4x cos 3x dx . Method 1(Integration by parts): Let u sin 4x and dv cos 3x dx du = 1 4 cos 4 x dx and v sin 3x . Thus, sin 4 x cos 3x dx 3 1 1 4 (sin 4 x) sin 3x cos 4 x sin 3x dx sin 4 x sin 3x 3 3 3 4 cos 4 x sin 3x dx . Find cos 4 x sin 3x dx . Let u cos 4x and dv = 3 1 sin 3x dx du 4 sin 4 x dx and v cos 3 x . Thus, 3 1 4 cos 4 x sin 3x dx cos 4 x cos 3x sin 4 x cos 3x dx . Returning to 3 3 1 the original integral, sin 4 x cos 3x dx = sin 4 x sin 3 x 3 4 1 4 1 sin 4 x cos 3x dx sin 4 x sin 3x cos 4 x cos 3x 3 3 3 3 4 16 7 cos 4 x cos 3x sin 4 x cos 3x dx sin 4 x cos 3x dx 9 9 9 1 4 sin 4 x sin 3x cos 4 x cos 3x sin 4 x cos 3x dx = 3 9 3 4 sin 4 x sin 3x cos 4 x cos 3x C . 7 7 Method 2(Trig identity): sin 4 x cos 3x dx 1 1 cos x cos 7 x C . 2 14 3 1 2 sin x sin 7x dx II. Integrating Powers of the Tangent and Secant Functions A. Useful trigonometric identity: tan 2 x 1 sec 2 x B. Useful integrals 1. sec x tan x dx sec x C 2. sec x dx tan x C 3. tan x dx ln sec x C ln cos x C 4. sec x dx ln sec x tan x C 2 C. Reduction formulas 1. 2. sec n 2 x tan x n 2 sec x dx sec n 2 x dx n 1 n 1 tan n 1 x tan x dx tan n 2 x dx n 1 n n D. Examples 1. Find 2 x dx . tan 2 x dx 2. Find tan (sec 2 x 1) dx sec 2 x dx 1dx tan x x C . tan xdx . 3 tan 3 xdx tan 2 x 2 tan x dx 1 tan 2 x ln sec x C . 2 4 3. Find sec xdx . 3 sec 3 x dx 4. Find sec x tan x 1 1 1 sec x dx sec x tan x ln sec x tan x C . 2 2 2 2 tan x sec x dx . 2 Let u tan x du sec 2 xdx tan x sec 2 x dx udu 1 2 u C 2 1 tan 2 x C . 2 5. Find tan x sec 4 x dx . tan x sec x dx tan x sec x sec x dx tan x(1 tan x) sec x dx tan x sec x dx tan x sec dx . Let u tan x du sec x dx . Thus, 1 1 1 1 tan x sec x dx udu u du u u C tan x tan x C . 2 4 2 4 4 2 2 3 2 2 2 2 4 6. Find 2 3 2 4 2 4 tan x sec 3 x dx . tan x sec x dx sec x (sec x tan x dx) . Let u sec x du sec x tan xdx . 1 1 Thus, tan x sec x dx u du u C sec x C . 3 3 3 2 3 7. Find 2 3 3 tan 2 x sec 3 x dx . tan x sec x dx (sec x 1) sec x dx sec x dx sec x dx . Using 1 3 the reduction formula, sec x dx sec tan x sec x dx . Thus, 4 4 2 3 2 3 5 5 3 5 3 3 1 3 sec 5 x dx sec 3 x dx sec 3 x tan x sec 3 xdx 4 4 1 1 1 1 sec 3 x dx sec 3 x tan x sec 3 x dx sec 3 x tan x sec x tan x 4 4 4 8 tan 2 x sec 3 x dx 1 ln sec x tan x C . 8 8. Find tan x sec 4 x dx . tan x sec 4 x dx tan x sec 2 x sec 2 x dx Let u tan x du sec 2 x dx tan x tan 2 x sec 2 x dx tan x (1 tan 2 x) sec 2 xdx . tan x sec 4 x dx tan x sec 2 x dx 5 1 2 3 2 7 u 2 du u 2 du u 2 u 2 C 3 7 3 7 2 2 (tan x) 2 (tan x) 2 C . 3 7 9. Find sec x tan x dx . Let u sec x u 2 sec x 2udu sec x tan xdx u 2 tan xdx 2udu 2 2 tan x dx 2 du . Thus, sec x tan x dx u du 2 1du u u u 2u C 2 sec x C . 6 Practice Sheet forTrigonometric Integrals (1) Prove the reduction formula: 1 n 1 sin n x dx sin n1 x cos x sin n2 x dx n n (2) Prove the reduction formula: cos n x dx (3) Prove the reduction formula: sec n 2 x tan x n 2 sec x dx sec n 2 x dx n 1 n 1 (4) Prove the reduction formula: tan n x dx (5) n 4 tan 3 (3 x ) dx = 0 (6) 4 cos 2 (2 x) dx = 0 8 sin( 5 x) cos(3x) dx = (8) tan 3 x sec 3 x dx = (9) sin x cos 3 x dx = (7) 1 n 1 cos n1 x sin x cos n2 x dx n n 0 7 tan n 1 x tan n 2 x dx n 1 (10) (11) cos 3 x sin 2 x dx = 2 sin 3 x cos x dx = 0 (12) sin 2 x cos 2 xdx (13) tan 5 x sec xdx Solution Key for Trigonometric Integrals (1) sin n x dx sin n1 x sin x dx . Use integration by parts with u sin n 1 x and dv sin x dx du (n 1) sin n2 x cos x dx and v sin x dx sin n n 1 sin x dx cos x x sin x dx = sin n1 x cos x (n 1) sin n2 x cos 2 x dx sin n1 x cos x (n 1) sin n2 x 1 sin 2 x dx sin n1 x cos x (n 1) sin n2 x dx (n 1) sin n x dx n sin n x dx sin n1 x cos x 1 n 1 (n 1) sin n2 x dx sin n x dx sin n1 x cos x sin n2 x dx . n n (2) cos x dx cos n n 1 x cos x dx . Use integration by parts with u cos n 1 x and 8 dv cos x dx du (n 1) cos n2 x ( sin x) dx and v cos x dx sin x cos n x dx cos n1 x cos x dx = cos n1 x sin x (n 1) cos n2 x sin 2 x dx cos n1 x sin x (n 1) cos n2 x 1 cos 2 x dx cos n1 x sin x (n 1) cos n2 x dx (n 1) cos n x dx n cos n x dx cos n1 x sin x (n 1) cos n2 x dx cos n x dx (3) 1 n 1 cos n1 x sin x cos n2 x dx . n n sec n x dx sec n2 x sec 2 x dx . Use integration by parts with u sec n 2 x and dv sec 2 x dx du (n 2) sec n3 x (sec x tan x dx) and v sec x dx sec n n2 sec 2 x dx tan x x sec 2 x dx = sec n2 x tan x (n 2) sec n2 x tan 2 x dx sec n2 x tan x (n 2) sec n2 x sec 2 x 1 dx sec n2 x tan x (n 2) sec n xdx (n 2) sec n2 x dx (n 1) sec n x dx sec n2 x tan x (n 2) sec n2 x dx (4) sec n x dx sec n 2 x tan x n 2 sec n 2 x dx . n 1 n 1 tan x dx tan n tan n 2 x dx n2 x tan 2 x dx tan n 1 x n 1 tan n2 tan n2 x dx . 9 x sec 2 x 1 dx tan n2 x sec 2 xdx (5) Let u 3x du 3 dx tan 3 (3x) dx reduction formula #4 above to get 1 1 tan 2 u ln sec u 6 3 4 1 1 tan 3 (3x) 3dx tan 3 u du . Use 3 3 1 1 tan 2 u 1 tan 3u du tan u du 3 3 2 3 1 1 4 tan (3 x ) dx = tan 2 (3x) ln sec(3x) 3 6 0 3 0 1 1 3 2 3 tan ln sec 4 3 4 6 1 2 1 1 1 (0) ln 1 ln 6 3 6 3 1 1 1 1 2 2 tan 0 ln sec0 (1) ln 2 3 3 6 6 2 . (6) Use the trigonometric identity cos 2 1 cos 2 to get 2 cos 2 (2 x) dx 1 cos(4 x) 1 1 1 1 dx 1dx cos(4 x) dx x sin 4 x 2 2 2 2 8 4 cos 2 (2 x) dx = 0 1 1 1 1 sin (0) sin( 0) . 8 8 2 4 8 2 1 (7) Use the trigonometric identity sin x cos y [sin( x y ) sin( x y )] to get 2 8 0 sin( 5x) cos(3x) dx 1 1 1 1 sin( 2 x) dx sin( 8x) dx cos( 2 x) cos(8x) 2 2 4 16 1 1 1 1 sin( 5 x) cos(3x) dx cos cos cos 0 cos 0 16 4 16 4 4 1 2 1 1 1 3 2 4 2 16 4 16 8 10 (8) (9) tan 3 x sec 3 x dx = (sec 2 x 1) sec 2 x (sec x tan x dx) sec 4 x (sec x tan x dx) 1 1 sec 2 x (sec x tan x dx) sec 5 x sec 3 x C . 5 3 (10) 1 5 2 cos x dx cos x sin cos 3 x sin 2 x dx = 2 2 1 2 1 sin xcos xdx 2 3 7 2 2 (sin x) 2 (sin x) 2 C . 3 7 x cos x dx 1 sin 2 x sin 2 x cos x dx sin 3 x cos x (cos x) dx 1 2 (cos x) 1 2 sin 2 x (sin x dx (cos x) 1 3 (sin x dx) (cos x) 2 (sin x dx) 2(cos x) 2 1 1 cos xsin x dx 2 2 5 2 (cos x) 2 5 5 5 8 2 2 2 dx = 2 cos cos 2 cos 0 cos 0 2 . 5 cos x 2 5 2 5 sin 3 x (12) Use the trigonometric identities cos 2 (sin x) 1 1 sin 2 x (cos x dx) sin 4 x (cos x dx) sin 3 x sin 5 x C . 3 5 2 0 sin x cos 2 x ( cos x dx) (sin x) 2 cos x dx (sin x) sin x cos 3 x dx = (11) tan 2 x sec 2 x (sec x tan x dx) sin 2 x cos 2 xdx 1 cos 2 1 cos 2 and sin 2 . 2 2 1 1 cos 2 x 1 cos 2 x dx 2 2 4 11 1 cos 2xdx 2 1 1 1 1 1 cos 4 x 1 1 1 dx cos 2 2 x dx x 1 dx dx x 4 4 4 4 2 4 8 1 1 1 1 1 1 cos 4 x dx x x sin 4 x C x sin 4 x C . 8 4 8 32 8 32 (13) tan 5 x sec xdx sec 2 tan 4 x tan x sec x dx 2 x 1 sec x tan x dx tan x 2 2 tan x sec x dx sec x 2 sec x 1sec x tan x dx 4 2 sec 4 x sec x tan x dx 2 sec 2 xsec x tan x dx sec x tan x dx 1 2 sec 5 x sec 3 x sec x C . 5 3 12