∫cos3(𝑥)sin(𝑥) d𝑥=

To solve the integral ∫cos3(𝑥)sin(𝑥) d𝑥, we can use the method of integration by parts. Here are the steps:

Step 1: Rewrite the integral We can rewrite the integral as ∫cos²(x)cos(x)sin(x) dx.

Step 2: Use the Pythagorean identity We can use the Pythagorean identity sin²(x) + cos²(x) = 1 to rewrite cos²(x) as 1 - sin²(x). So, the integral becomes ∫(1 - sin²(x))cos(x)sin(x) dx.

Step 3: Use substitution Let u = sin(x), then du = cos(x) dx. The integral becomes ∫(1 - u²)u du.

Step 4: Expand and integrate Expand the integral to ∫u - u³ du. Now, we can integrate term by term to get 1/2u² - 1/4u⁴ + C.

Step 5: Substitute back Finally, substitute u = sin(x) back into the equation to get the final answer: 1/2sin²(x) - 1/4sin⁴(x) + C.