Step 1: Rewrite the integral by taking out a sin (x):

ò (sin³(x) cos²(x))dx = ((sin²(x) cos²(x) sin(x))dx

Step 2: Substitute sin²(x) = 1 - cos²(x) into the integral:

ò ((sin²(x) cos²(x) sin(x))dx = (1- cos²(x))(cos²(x))(sin x)dx

Step 3: Multiply through and rewrite:

ò (1- cos²(x))(cos²(x))(sin x)dx = (cos²(x) - cos⁴(x))dx

Step 4: Substitute u = cos x and du = -sin x dx into the integral:

ò (cos²(x) - cos⁴(x))dx = - (u² - u⁴)du

Step 5: Separate into two integrals by using the Sum Rule:

- ò (u² - u⁴)du = - ò u² du + ò u⁴ du

Step 6: Integrate both integrals using the Power Rule:

- òu² du + ò u⁴ du = - u^{(2 + 1)} + u^{(4 + 1)} + C = - u³ + u⁵ + C

Step 7: Substitute the original values into the equation:

- u³ + u⁵ + C = cos⁵(x) - cos³(x) + C.