Final answer:
To evaluate the integral ∫(6sin²(x)cos³(x))dx, we can use the power-reduction formula for cosine and the substitution method to simplify and solve the integral. The final answer is (3/8)sin(2x) - (1/8)sin³(2x) + C.
Step-by-step explanation:
To evaluate the integral ∫(6sin²(x)cos³(x))dx, we can use the power-reduction formula for cosine: cos²(x) = (1 + cos(2x))/2. Applying this formula, we have:
∫(6sin²(x)cos³(x))dx = 6∫(sin²(x)(1 + cos(2x))/2*cos³(x))dx
Using the identity sin²(x) = (1 - cos(2x))/2, we can further simplify:
6∫(((1 - cos(2x))/2)(1 + cos(2x))/2*cos³(x))dx
Expanding and simplifying the integrand, we have:
6∫((1/4)(1 - cos²(2x))*cos³(x))dx
Now, we can use the substitution u = sin(2x), du = 2cos(2x)dx:
6(1/4)∫(1 - u²)du
Applying the power rule for integration, we get:
6(1/4)(u - u³/3) + C
Substituting u = sin(2x) back in, we have:
6(1/4)(sin(2x) - sin³(2x)/3) + C
Simplifying further:
(3/8)sin(2x) - (1/8)sin³(2x) + C