4. sin3 (2x +1)
The given function is sin 3 ( 2 x + 1 )
∫ sin 3 ( 2 x + 1 ) d x (1)
Also, sin 3 y = 3 sin y − 4 sin 3 y sin 3 y = 1 4 ( 3 sin y − sin 3 y ) (2)
Compare (1) and (2), we get,
y = 2 x + 1
Substitute value of y in (2).
sin 2 ( 2 x + 1 ) = 3 4 sin ( 2 x + 1 ) − 1 4 sin ( 2 x + 1 )
Substitute value of sin 3 ( 2 x + 1 ) in (1) to determine the integration of the function.
∫ sin 3 ( 2 x + 1 ) d x = 3 4 ∫ sin ( 2 x + 1 ) d x − 1 4 ∫ sin ( 6 x + 3 ) = − 3 8 cos ( 2 x + 1 ) + 1 24 cos ( 6 x + 3 )
Thus, the integral of the function sin 3 ( 2 x + 1 ) is − 3 8 cos ( 2 x + 1 ) + 1 24 cos ( 6 x + 3 ) .
The given function is
Also,
Compare (1) and (2), we get,
Substitute value of
Substitute value of
Thus, the integral of the function
