EXAMPLE 2 Evaluate cos(𝜃)sin2(𝜃) d𝜃.SOLUTION This indefinite integral isn't immediately apparent in the Table of Indefinite Integrals, so we use trigonometric identities to rewrite the function before integrating:cos(𝜃)sin2(𝜃) d𝜃 =  1sin(𝜃)cos(𝜃)  d𝜃 =  csc(𝜃) cot(𝜃) d𝜃 =  + C.

The text you provided seems to be a bit mixed up. However, I can provide a step-by-step solution to the integral of cos(𝜃)sin^2(𝜃) d𝜃.

Step 1: Use the power-reducing identity sin^2(𝜃) = 1/2 - 1/2cos(2𝜃). The integral becomes ∫ cos(𝜃)(1/2 - 1/2cos(2𝜃)) d𝜃.

Step 2: Distribute the cos(𝜃) to get 1/2∫ cos(𝜃) d𝜃 - 1/2∫ cos(𝜃)cos(2𝜃) d𝜃.

Step 3: The first integral can be solved directly to get 1/2sin(𝜃). The second integral can be solved using the product-to-sum identity cos(𝜃)cos(2𝜃) = 1/2[cos(𝜃 - 2𝜃) + cos(𝜃 + 2𝜃)] = 1/2[cos(-𝜃) + cos(3𝜃)].

Step 4: Integrate to get -1/2sin(𝜃) + 1/6sin(3𝜃).

Step 5: Combine the results from step 3 and 4 to get the final answer: 1/2sin(𝜃) - 1/2sin(𝜃) + 1/6sin(3𝜃) + C = 1/6sin(3𝜃) + C.