INTEGRATION BY DECOMPOSITION METHOD EXAMPLES

Integrate the following functions with respect to x :

Example 1 :

cos2x/(sin²x cos²x)

Solution :

= ∫[cos2x/(sin²x cos²x)]dx

= ∫[(cos²x - sin²x)/(sin²x cos²x)]dx

= ∫[cos²x/(sin²x cos²x)]dx - ∫[sin²x/(sin²x cos²x)]dx

= ∫(1/sin²x)dx - ∫(1/cos²x)dx

= ∫cosec²xdx - ∫sec²xdx

= -cotx - tanx + c

Example 2 :

(3 + 4cosx)/sin²x

Solution :

= ∫[(3 + 4 cosx)/sin²x]dx

  = 3∫(1/sin²x)dx + 4∫(cos x/sin²x)dx

= 3∫cosec²xdx + 4∫cotxcosecxdx

= 3(-cotx) + 4(-cosecx) + c

= -3cotx - 4cosecx + c

Example 3 :

sin²x/(1 + cosx)

Solution :

= ∫[sin²x/(1 + cosx)]dx

= ∫[(1 - cos²x)/(1 + cosx)]dx

= ∫[(1 + cosx)(1 - cosx)/(1 + cosx)]dx

= ∫(1 - cosx)dx

= x - sinx + c

Example 4 :

sin4x/sinx

Solution :

= ∫[sin4x/sinx]dx

= ∫[sin2(2x)/sinx]dx

= ∫[2sin(2x)cos(2x)/sinx]dx

= ∫[2 ⋅ 2sinxcosx ⋅ cos(2x)/sinx]dx

= ∫[2 ⋅ 2cosx ⋅ cos(2x)]dx

= ∫2[cos(x + 2x) + cos(-x)]dx

= 2∫[cos3x + cosx]dx

= 2[(sin3x)/3 + (sinx)] + c

Example 5 :

cos3xcos2x

Solution :

= ∫[cos3xcos2x]dx

= ∫(2/2)[cos3xcos2x]dx

= (1/2)∫(2cos3xcos2x)dx

= (1/2)∫[(cos(3x + 2x) + cos(3x - 2x)]dx

= (1/2)∫[cos5x + cosx]dx

= (1/2)[(sin5x)/5 + sinx] + c

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