7.sin 4x sin 8x
The given function is sin 4 x sin 8 x
∫ sin 4 x sin 8 x d x (1)
Also, sin A sin B = 1 2 { cos ( A − B ) − cos ( A + B ) } (2)
Compare (1) and (2),
A = 4 x and B = 8 x
Use (2) to further solve the equations,
∫ sin 4 x sin 8 x d x = ∫ cos ( 4 x − 8 x ) − cos ( 4 x + 8 x ) d x = 1 2 ∫ { cos ( − 4 x ) − cos ( 12 x ) } d x = 1 2 ∫ cos ( 4 x ) d x − 1 2 ∫ cos ( 12 x ) d x = 1 2 ( sin 4 x 4 − sin 12 x 12 ) + c
Thus, the integral of the function sin 4 x sin 8 x is 1 2 ( sin 4 x 4 − sin 12 x 12 ) + c .
The given function is
Also,
Compare (1) and (2),
Use (2) to further solve the equations,
Thus, the integral of the function
