Evaluate the integral ∫(sin3x)(cos3x)dx

1. The task at hand involves the evaluation of the integral represented by the expression ∫(sin(3x))(cos(3x))dx. This integral requires a methodical approach to simplify the integrand before proceeding with the integration process. One effective technique to tackle this integral is to utilise a trigonometric identity that relates the product of sine and cosine functions to a single sine function. Specifically, the identity sin(A)cos(A) = 1/2 sin(2A) can be employed, where A is the angle in question.

2. By applying this identity to the integral, we can rewrite the integrand as follows: sin(3x)cos(3x) becomes (1/2)sin(6x). Consequently, the integral can be transformed into a more manageable form: ∫(1/2)sin(6x)dx. This simplification allows for a straightforward integration process, as the integral of sin(kx) is known to be (-1/k)cos(kx) plus a constant of integration. In this case, k is equal to 6, which will facilitate the calculation.

3. Upon integrating the simplified expression, we find that the integral ∫(1/2)sin(6x)dx results in (-1/12)cos(6x) plus a constant of integration, C. Therefore, the final result of the original integral ∫(sin(3x))(cos(3x))dx can be expressed as (-1/12)cos(6x) + C. This solution not only provides the value of the integral but also illustrates the effectiveness of using trigonometric identities to simplify complex integrals in calculus.