Question

How do I find the indefinite integral of sin(3x)cos(3x)?

a) 1/3sin²(3x)+C
b) -1/6cos²(3x)+C
c) -1/6sin²(3x)+C
d) 1/6cos²(3x)+C

Answer 1

The indefinite integral of sin(3x)cos(3x) is (2/3)(sin(6x))^(3/2) + C.

Step-by-step explanation:

To find the indefinite integral of sin(3x)cos(3x), we can use the double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ). Let θ = 3x. So, the integral becomes ∫ sin(3x)cos(3x) dx = ∫ (2sin(3x)cos(3x))(1/2) dx. Using the double angle formula, we have ∫ sin(3x)cos(3x) dx = ∫ (sin(6x))(1/2) dx.

Now, using the power rule for integration, the integral of (sin(6x))(1/2) dx is (2/3)(sin(6x))^(3/2) + C, where C is the constant of integration. Hence, the indefinite integral of sin(3x)cos(3x) is (2/3)(sin(6x))(3/2) + C.