Question Detail
Question
Using the reduction formula, evaluate ∫ cos⁴x dx.
Correct Answer
Option D is the correct answer.
Detailed Explanation
The reduction formula for ∫ cosⁿx dx is given by ∫ cosⁿx dx = (1/n)cosⁿ⁻¹x sinx + ((n-1)/n) ∫ cosⁿ⁻²x dx.First, apply the formula for n=4:∫ cos⁴x dx = (1/4)cos³x sinx + (3/4) ∫ cos²x dxNext, apply the formula for n=2 to ∫ cos²x dx:∫ cos²x dx = (1/2)cos¹x sinx + (1/2) ∫ cos⁰x dx= (1/2)cos x sinx + (1/2) ∫ 1 dx= (1/2)cos x sinx + (1/2)x + C'Substitute this back into the first expression:∫ cos⁴x dx = (1/4)cos³x sinx + (3/4) [(1/2)cos x sinx + (1/2)x] + C= (1/4)cos³x sinx + (3/8)cos x sinx + (3/8)x + C
Hint
Recall and apply the reduction formula for ∫ cosⁿx dx twice. The formula is: ∫ cosⁿx dx = (1/n)cosⁿ⁻¹x sinx + ((n-1)/n) ∫ cosⁿ⁻²x dx.