Final answer:
The integral ∫7sin^4(x)cos^2(x)dx can be evaluated using double angle formulas, converting cos^2(x) and sin^4(x) into terms involving cos(2x) and then integrating term by term.
Step-by-step explanation:
To find the integral ∫7sin4(x)cos2(x)dx using a double angle formula, we can use the identities for sine and cosine functions. Specifically, we can use the identity cos2θ = ½(1 + cos(2θ)) to simplify the integral. First, rewrite cos2(x) as ½(1 + cos(2x)). Next, for sin4(x), express it as (sin2(x))2 and then use the double angle formula sin2(x) = ½(1 - cos(2x)). Substituting these into the integral, we obtain
∫7(½(1 - cos(2x)))2(½(1 + cos(2x)))dx
By expanding and simplifying the integrand, you can integrate term by term, possibly using further trigonometric identities or substitution if needed, to find the final result.