Evaluate the integral sin4x cos^2 4x dx
A. cos^3 4x/3 +C
B. - cos^3 4x/3 +C
C. cos^3 4x/12 +C
D. -cos^3 4x/12 +C
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Notice that the derivative of the power cos^2 (4x) shows up as multiple, so we can just reverse the chain rule
I know that it must be something like (cos 4x)^3....
so from y ' = sin 4x (cos 4x)^2
we get y = sin 4x (cos 4x)^3 (1/3) (-1/sin 4x) (1/4)
= (-1/12) cos^3 (4x) )
which if you had used brackets correctly would have been C>
To evaluate the integral ∫ sin^4(x) cos^2(x) dx, we can use the trigonometric identity:
sin^2(x) = 1/2 - (1/2)cos(2x)
We can rewrite sin^4(x) as (sin^2(x))^2 and use the identity to simplify it further:
(sin^2(x))^2 = (1/2 - (1/2)cos(2x))^2
Now, let's expand and simplify this expression:
(1/2 - (1/2)cos(2x))^2 = (1/2)^2 - 2(1/2)(1/2)cos(2x) + ((1/2)cos(2x))^2
= 1/4 - 1/2cos(2x) + 1/4cos^2(2x)
= 1/4 - 1/2cos(2x) + 1/4(1/2 + 1/2cos(4x))
= 1/4 - 1/2cos(2x) + 1/8 + 1/8cos(4x)
= 3/8 - 1/2cos(2x) + 1/8cos(4x)
Now, we can rewrite the original integral as:
∫ sin^4(x) cos^2(x) dx = ∫ (3/8 - 1/2cos(2x) + 1/8cos(4x))cos^2(x) dx
Expanding this expression, we get:
∫ (3/8)cos^2(x) dx - ∫ (1/2)cos(2x)cos^2(x) dx + ∫ (1/8)cos(4x)cos^2(x) dx
Using the power reduction formulas, we can simplify each integral:
∫ (3/8)cos^2(x) dx = (3/8)∫ (1 + cos(2x))/2 dx
= (3/8)(x/2 + (1/4)sin(2x)) + C1
∫ (1/2)cos(2x)cos^2(x) dx = (1/2)∫ (cos(2x)/2)(1 + cos(2x)) dx
= (1/4)∫ (1 + cos(2x))^2 dx
= (1/4)∫ (1 + 2cos(2x) + cos^2(2x)) dx
= (1/4)[x + (1/2)sin(2x) + (1/4)(x/2 + (1/4)sin(4x))] + C2
∫ (1/8)cos(4x)cos^2(x) dx = (1/8)∫ (cos(4x)/2)(1 + cos(2x)) dx
= (1/16)∫ (1 + cos(4x) + cos^2(2x) + cos(6x)) dx
= (1/16)[x + (1/4)sin(4x) + (1/4)(x/2 + (1/4)sin(2x)) + (1/6)(x/2 + (1/4)sin(6x))] + C3
Now, adding all the results together, we get:
∫ sin^4(x) cos^2(x) dx = (3/8)(x/2 + (1/4)sin(2x)) - (1/4)[x + (1/2)sin(2x) + (1/4)(x/2 + (1/4)sin(4x))] + (1/16)[x + (1/4)sin(4x) + (1/4)(x/2 + (1/4)sin(2x)) + (1/6)(x/2 + (1/4)sin(6x))] + C
Simplifying this expression, we obtain:
∫ sin^4(x) cos^2(x) dx = (3/8)x/2 + (1/32)sin(2x) + (1/128)sin(4x) + (1/144)sin(6x) + C
Thus, none of the given answer choices match the result obtained.