Question Detail
Question
Find ∫ sin³ x cos³ x dx.
Correct Answer
Option A is the correct answer.
Detailed Explanation
To integrate ∫ sin³ x cos³ x dx, we can rewrite one of the odd powers using a Pythagorean identity. Let's express sin³ x as sin² x · sin x and use sin² x = 1 - cos² x. The integral becomes ∫ (1 - cos² x) cos³ x sin x dx. Now, use u-substitution. Let u = cos x. Then du = -sin x dx, so sin x dx = -du. Substitute these into the integral: ∫ (1 - u²) u³ (-du) = ∫ -(u³ - u⁵) du = ∫ (u⁵ - u³) du. Integrate term by term: (u⁶/6) - (u⁴/4) + C. Substitute back u = cos x: (cos⁶ x)/6 - (cos⁴ x)/4 + C.
Hint
When dealing with odd powers of sine and cosine, save one factor of the odd power and convert the remaining even power using the identity sin² x + cos² x = 1. Then, use u-substitution with u being the other trigonometric function.