Question Detail
Question
Evaluate ∫ sin 4x cos 2x dx.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Use the product-to-sum trigonometric identity: sin A cos B = (1/2) [sin(A+B) + sin(A-B)]. Here, A = 4x and B = 2x. So, sin 4x cos 2x = (1/2) [sin(4x+2x) + sin(4x-2x)] = (1/2) [sin 6x + sin 2x]. Now integrate this expression: ∫ (1/2) [sin 6x + sin 2x] dx = (1/2) [∫ sin 6x dx + ∫ sin 2x dx]. The integral of sin(ax) is -(1/a)cos(ax). So, (1/2) [-(1/6)cos 6x - (1/2)cos 2x] + C = -(1/12)cos 6x - (1/4)cos 2x + C. Rearranging to match the option format, we get -(1/4)cos 2x - (1/12)cos 6x + C.
Hint
Transform the product of sine and cosine into a sum using the identity sin A cos B = (1/2)[sin(A+B) + sin(A-B)] before integrating.