Question Detail
Question
By Wallis' formula, evaluate ∫0π/2 cos5 x sin2 x dx.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Wallis' formula for ∫0π/2 sinm x cosn x dx is given by:[(m-1)!! (n-1)!!] / [(m+n)!!] * Kwhere K = π/2 if both m and n are even, and K = 1 otherwise.In this case, m=2 (for sin² x) and n=5 (for cos⁵ x).(m-1)!! = (2-1)!! = 1!! = 1.(n-1)!! = (5-1)!! = 4!! = 4 × 2 = 8.(m+n)!! = (2+5)!! = 7!! = 7 × 5 × 3 × 1 = 105.Since one exponent (5) is odd, K = 1.So the integral value is (1 × 8) / 105 × 1 = 8/105.
Hint
Apply Wallis' formula: [(m-1)!! (n-1)!!] / [(m+n)!!]. Remember to check if m and n are both even for the π/2 factor.