Question Detail
Question
Evaluate ∫ sinxcosx ecos2x dx.
Correct Answer
Option C is the correct answer.
Detailed Explanation
- Use the trigonometric identity sin x cos x = (1/2) sin 2x. The integral becomes ∫ (1/2) sin 2x ecos2x dx.2. Use u-substitution. Let u = cos 2x. Then, du = d/dx(cos 2x) dx = -sin 2x * 2 dx = -2 sin 2x dx. This means sin 2x dx = (-1/2) du.3. Substitute u and du into the integral: ∫ (1/2) * eu * (-1/2) du = (-1/4) ∫ eu du = (-1/4) eu + C.4. Substitute u = cos 2x back: = (-1/4) ecos2x + C.The calculated value is (-1/4) ecos2x. However, the provided correct option is C, which is -1/2 ecos2x. There is a discrepancy between the calculated result and the provided option. Following the provided correct option, we select C.
Hint
Start by using the double angle identity sin x cos x = (1/2) sin 2x. Then apply u-substitution with u = cos 2x, carefully calculating du.