Evaluate the integral ∫(sin3x)(cos3x)dx (C4 Integration)

First, we must recognise that the integral is written as a product of two functions which cannot be directly integrated, therefore a trigonometric identity must be used to express this a single function. Since both functions have the same value within their brackets (3x) and is the product of sine and cosine, we can use the sine double angle formula to express the integral in terms of sine only. Recall that the double angle formula for sine is; sin(2x)=2sin(x)cos(x). Therefore sin(3x)cos(3x) can be written as (1/2)sin(6x).This is then a simple trig integral using the reverse chain rule and remembering that the integral of sine is -cosine; ∫(1/2)sin(6x)dx= -(1/12)cos(6x) + c. Since this is an indefinite integral, we must remember to add the arbitrary constant, c, onto the end. We can always check to see if this is correct by differentiating our answer to see if we get the initial integral.