How do you integrate sin 3x cos 3xdx?

To solve the integral of sin(3x) cos(3x) dx, we can use a trigonometric identity to simplify the expression.

Step 1: Use the trigonometric identity
We use the following trigonometric identity to simplify the integrand:

sin(A) cos(A) = 1/2 sin(2A)

Here, A is 3x, so we can apply the identity as follows:

sin(3x) cos(3x) = 1/2 sin(6x)

Step 2: Set up the integral
Now the integral becomes:

∫ sin(3x) cos(3x) dx = ∫ (1/2) sin(6x) dx

Step 3: Integrate the expression
We now integrate the expression (1/2) sin(6x) with respect to x.

The integral of sin(kx) with respect to x is -1/k cos(kx). In this case, k = 6.

Thus:

∫ (1/2) sin(6x) dx = (1/2) * (-1/6) cos(6x)

Step 4: Simplify the result
Now, simplify the result:

(1/2) * (-1/6) cos(6x) = -1/12 cos(6x)

Step 5: Add the constant of integration
Finally, we add the constant of integration, C:

The solution is:

-1/12 cos(6x) + C