Grade 12Integral Calculus

Integrate the following ...........,,....,: sin4x.cos3x.sin2x

Ritz

8 Years agoGrade 12

1 Answer

Arun

Approved Tutor Answer8 Years ago

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)