Final answer:
To evaluate the integral of sin²(x) cos(x) from 0 to π, we can rewrite the expression using trigonometric identities and then evaluate it step by step. The final answer is (6π - 2√3 - 3)/24, which simplifies to 3/4.
Step-by-step explanation:
To evaluate the integral of sin²(x) cos(x) from 0 to π, we can use the identity cos²(x) = 1 - sin²(x) to rewrite the integral as ∫sin²(x)(1 - sin²(x))dx. Expanding the expression gives us ∫sin²(x) - sin⁴(x) dx. We can then use the power reducing identity sin²(x) = (1-cos(2x))/2 to simplify the integral to ∫(1-cos(2x))/2 - sin⁴(x) dx. Using linearity of the integral, we can split it into two integrals: ∫(1-cos(2x))/2 dx - ∫sin⁴(x) dx. Evaluating each integral gives us (x/2) - (1/4)sin(2x) - (√3x)/12 - (1/8)sin(4x). Now we can evaluate this expression from 0 to π. Plugging in the values and simplifying, we get π/2 - (√3π)/12 - 1/8 = (6π - 2√3 - 3)/24. Therefore, the correct answer is d) 3/4.