Example 1 :
Integrate the following functions with respect to x :
sin2x / (1 + cos x)
Solution :
∫[sin2x / (1 + cos x)] dx
= ∫[(1 - cos2x) / (1 + cos x)] dx
= ∫[(1 + cos x)(1 - cosx) / (1 + cos x)] dx
= ∫(1 - cos x) dx
= x - sin x + c
Example 2 :
Integrate the following functions with respect to x :
sin 4x/sin x
Solution :
∫[sin 4x/sin x] dx
sin 4x/sin x = sin 2 (2x) / sin x
= 2 sin 2x cos 2x / sin x
= 2 (2 sinx cos x) cos 2x / sin x
= 4 cos x cos 2x
= 2 (2 cos x cos 2x)
= 2 [cos (x + 2x) + cos (-x)]
sin 4x/sin x = 2 [cos 3x + cos x]
= 2 ∫ [cos 3x + cos x] dx
= 2 [(sin 3x)/3 + (sin x)] + c
Example 3 :
Integrate the following functions with respect to x :
cos 3x cos 2x
Solution :
∫[cos 3x cos 2x] dx
= ∫(2/2) [cos 3x cos 2x] dx
= (1/2) ∫(2cos 3x cos 2x) dx
= (1/2) ∫[(cos (3x + 2x) + cos (3x - 2x)] dx
= (1/2) ∫[cos 5x + cos x] dx
= (1/2) [(sin 5x)/5 + sin x] + c
Example 4 :
Integrate the following functions with respect to x :
sin2 5x
Solution :
∫[sin2 5x] dx
= ∫[1 - cos 2(5x)] dx
= ∫[1 - cos 10x] dx
= x - sin 10x /10 + c
= x - (1/10) sin 10x + c
Eample 5 :
Integrate the following functions with respect to x :
(1 + cos 4x) / (cot x - tan x)
Solution :
∫[(1 + cos 4x) / (cot x - tan x)] dx
cos 2x = 2cos2x - 1
2cos2x = 1 + cos 2x
2cos22x = 1 + cos 4x
= (2cos22x) / ((cos x/sin x) - (sin x/cos x))
= (2cos22x) / ((cos2x - sin2x)/(sin x cos x))
= (2cos22x) ⋅ ((sin x cos x) / cos 2x)
= 2 cos 2x sin x cos x
= cos 2x (2 sin x cos x)
= cos 2x sin 2x
= ∫(2/2) (cos 2x sin 2x) dx
= (1/2) ∫ sin 2(2x) dx
= (1/2) ∫sin 4x dx
= -(1/2) (cos 4x)/4 + c
= -(1/8) (cos 4x) + c
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