11. cos4 2x
The given function is cos 4 2 x ,
∫ cos 4 x d x = ∫ ( cos 2 2 x ) 2 d x (1)
Also, cos 2 x = ( 1 + cos ( 2 x ) 2 )
From (1),
cos 4 x = ( 1 + cos ( 2 x ) 2 ) 2 = 1 4 ( 1 + cos 2 4 x + 2 cos 4 x ) = 1 4 [ 1 + ( 1 + cos ( 8 x ) 2 ) + 2 cos 4 x ] = 1 4 [ 3 2 + 1 2 cos ( 8 x ) + 2 cos 4 x ] (2)
From (1) and (2), we get,
∫ cos 4 x d x = ∫ 1 4 [ 3 2 + 1 2 cos ( 8 x ) + 2 cos 4 x ] d x = ∫ ( 3 8 + 1 8 cos ( 8 x ) + 1 2 cos 4 x ) d x = 3 x 8 + sin 8 x 64 + 1 8 sin ( 4 x ) + c
Thus, the integral of the function cos 4 x is 3 x 8 + sin 8 x 64 + 1 8 sin ( 4 x ) + c .
The given function is
Also,
From (1),
From (1) and (2), we get,
Thus, the integral of the function
