What is the integral of sin4xcos4x dx?
To find the integral of sin(4x)cos(4x) dx, you can use a trigonometric identity and integration techniques.
First, we can use the double angle identity for sine:
sin(4x) = 2sin(2x)cos(2x).
Now, the integral becomes:
∫ (2sin(2x)cos(2x))cos(4x) dx.
Next, we use the product-to-sum identity for cosine:
cos(2x)cos(4x) = (1/2)(cos(2x + 4x) + cos(2x - 4x)).
Simplifying further, the integral becomes:
∫ [(1/2)(2sin(2x)cos(6x) + 2cos(2x)cos(-2x))] dx.
Now, we can integrate each term separately:
∫ [sin(2x)cos(6x) + cos(2x)cos(-2x)] dx.
The integral of sin(2x)cos(6x) can be found using the substitution method or integration by parts.
For simplicity, let's focus on the second term, cos(2x)cos(-2x).
Using the identity cos(-θ) = cos(θ), we simplify it to:
∫ cos²(2x) dx.
Now, we use the trigonometric identity:
cos²(2x) = (1/2)(1 + cos(4x)).
The integral becomes:
∫ [(1/2)(1 + cos(4x))] dx.
Expanding and integrating, we get:
(1/2)∫dx + (1/2)∫cos(4x) dx.
The integral of dx is x, and the integral of cos(4x) can be found by dividing the angle by the coefficient and using the sine function:
(1/2)(x) + (1/2)(1/4)sin(4x) + C,
where C is the constant of integration.
Therefore, the final answer to the integral of sin(4x)cos(4x) dx is:
(1/2)x + (1/8)sin(4x) + C.