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integral tak tentu fungsi trigonometri?
tentukan hasil dari setiap integral:
1. integral (tan(2x-1)-sec(2x-1))^2 dx
2. integral (sin^3 x+ 3 cos x)/2 sin^2 x dx
3. integral (tan^2 x + cos t^2 x + 4) dx
4. integral tan^2 3x - cotan^2 2x dx
5. integral sin x - cos 2x dx
3 Answers
- 9 years agoFavorite Answer
1. ∫ (tan (2x-1) - sec (2x-1))² dx
= ∫ (tan² (2x-1) - 2 tan (2x-1) sec (2x-1) + sec² (2x-1)) dx
= ∫ ((sec² (2x-1) - 1) - 2 tan (2x-1) sec (2x-1) + sec² (2x-1)) dx
= ((½ tan (2x-1) - x) - 2(½ sec (2x-1)) + ½ tan (2x-1)) + c
= (½ tan (2x-1) - x - sec (2x-1) + ½ tan (2x-1)) + c
= tan (2x-1) - sec (2x-1) - x + c
2. ∫ ((sin^3 x+ 3 cos x)/(2 sin² x)) dx
= ∫ ((sin x (sin² x) + 3 cos x)/(1 - cos 2x)) dx
= ∫ ((sin x (½ - ½ cos 2x) + 3 cos x)/(1 - cos 2x)) dx
= ∫ ((½ sin x - ½ sin x cos 2x) + 3 cos x)/(1 - cos 2x)) dx
= ∫ ((½ sin x - ¼(2 cos 2x sin x)) + 3 cos x)/(1 - cos 2x)) dx
= ∫ ((½ sin x - ¼(sin 3x - sin x)) + 3 cos x)/(1 - cos 2x)) dx
= ∫ ((½ sin x - ¼ sin 3x + ¼ sin x) + 3 cos x)/(1 - cos 2x)) dx
= ((-½ cos x + (1/12) cos 3x - ¼ cos x) + 3 sin x)/(x - ½ sin 2x)) + c
= ((-3/4) cos x + (1/12) cos 3x + 3 sin x)/(x - ½ sin 2x)) + c
3. ∫ (tan² x + cos² x + 4) dx
= ∫ ((sec² x - 1) + (½ + ½ cos 2x) + 4) dx
= ((tan x - x) + (½x + ¼ sin 2x) + 4x) + c
= tan x - ¼ sin 2x + (7/2)x + c
4. ∫ (tan² 3x - cotan² 2x) dx
= ∫ ((sec² 3x - 1) - (cosec² 2x - 1)) dx
= (((1/3) tan 3x - x) - (-½ cotan 2x - x)) + c
= ((1/3) tan 3x - x + ½ cotan 2x + x) + c
= (1/3) tan 3x + ½ cotan 2x + c
5. ∫ (sin x - cos 2x) dx
= -cos x - ½ sin 2x + c
- Anonymous7 years ago
∫sin 4x cos 4x dx ?
