Answer:
We can start by simplifying the expression inside the parentheses using the identity:
cos 2x = 2 cos² x - 1
Substituting this in, we get:
1 – 3 cos 2x + 3 cos² 2x − cos³ 2x
= 1 – 3(2 cos² x - 1) + 3(2 cos² x - 1)² − (2 cos² x - 1)³
= 1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x
Therefore, we can rewrite f(x) as:
f(x) = 4 cos x (1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x)
Next, we can use the trigonometric identity:
sin 2x = 2 cos x sin x
to express cos x in terms of sin x:
cos x = √(1 - sin² x)
Substituting this in, we get:
f(x) = 4 sin x cos³ x (1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x)
= 4 sin x (√(1 - sin² x))³ (1 – 6 (2 sin² x - 1) + 9 (2 sin² x - 1)² - 4 (2 sin² x - 1)³)
= 4 sin x (1 - sin² x)^(3/2) (16 sin⁶ x - 48 sin⁴ x + 36 sin² x - 8)
Next, we can use the substitution u = 1 - sin² x, du = -2 sin x cos x dx, to obtain:
f(x) dx = -2 du (u^(3/2)) (16 - 48u + 36u² - 8u³)
Integrating, we get:
f(x) dx = 2/3 (1 - sin² x)^(5/2) (8 - 36(1 - sin² x) + 36(1 - sin² x)² - 8(1 - sin² x)³) + C
Now, we can use the trigonometric identity:
sin² x = (1 - cos 2x)/2
to simplify the expression inside the parentheses. After some algebra, we obtain:
f(x) dx = 3/2 sin 7x + C
where C is the constant of integration. Since m is a positive real constant, we can set:
7x = m
and solve for x:
x = m/7
Substituting this in, we get:
f(x) dx = 3/2 sin(7m/7) = 3/2 sin m
Therefore, we have shown that:
f(x) dx = 3/2 sin m, where m is a positive real constant.