Final answer:
To evaluate the integral ∫sin^3 (θ)cos^4 (θ)dθ, use trigonometric identities to rewrite the expression and then split it into two separate integrals. The first integral can be evaluated using the power rule, while the second integral requires a substitution or integration by parts.
Step-by-step explanation:
To evaluate the integral ∫sin^3 (θ)cos^4 (θ)dθ, we will use trigonometric identities to simplify the expression. Let's start by using the identity sin^2 (θ) + cos^2 (θ) = 1 to rewrite sin^3 (θ) as sin (θ)sin^2 (θ). Next, we will use the identity sin^2 (θ) = 1 - cos^2 (θ) to substitute sin^2 (θ) in the expression. Substituting these identities, we get ∫sin (θ)(1 - cos^2 (θ))cos^4 (θ)dθ.
Expanding the expression, we have ∫sin (θ)cos^4 (θ) - sin (θ)cos^6 (θ)dθ. We can now split the integral into two separate integrals: ∫sin (θ)cos^4 (θ)dθ - ∫sin (θ)cos^6 (θ)dθ. The first integral can be easily evaluated using the power rule for integrals, while the second integral can be reduced using a substitution or integration by parts method.