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Grade: 12
        Integrate  the following ...........,,....,: sin4x.cos3x.sin2x
7 months ago

Answers : (1)

Arun
12289 Points
							
Note that sin(A)sin(B) = (1/2) [cos(A - B) - cos(A + B)]. 
Also, cos(-A) = cos(A). 

Therefore, 
sin(2x)sin(4x) = (1/2) [cos(2x) - cos(6x)]. 

Also note that cos(A)cos(B) = (1/2) [cos(A + B) + cos(A - B)]. 

∫sin(2x)cos(3x)sin(4x) dx 
= ∫ (1/2) [cos(2x) - cos(6x)] cos(3x) dx 
= (1/2) ∫ [cos(2x)cos(3x) - cos(6x)cos(3x)] dx 
= (1/2)(1/2) ∫ [cos(5x) + cos(x) - cos(9x) - cos(3x)] dx 
= (1/4) ∫[cos(x) - cos(3x) + cos(5x) - cos(9x)] dx 
= (1/4) [ sin(x) - sin(3x)/3 + sin(5x)/5 - sin(9x)/9 ]
 
 
Regards
Arun (askIITians forum expert)
7 months ago
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