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 (even) and n = 5 (odd). Since n is odd, K = 1.m-1 = 1, so (m-1)!! = 1!! = 1.n-1 = 4, so (n-1)!! = 4!! = 4 × 2 = 8.m+n = 2+5 = 7, so (m+n)!! = 7!! = 7 × 5 × 3 × 1 = 105.Substituting these values into the formula:(1 × 8) / 105 = 8/105.
Hint
Apply Wallis' formula. Remember to correctly identify m and n, calculate the double factorials, and determine the value of K (π/2 or 1).