INTEGRATION OF TRIGONOMETRIC FUNCTIONS EXAMPLES

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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