This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To solve the integral _0^/2 8 ^4 y ^2 y\,dy, we will use trigonometric identities to reduce the powers of sine and cosine. Step 1: Rewrite the integrand using half-angle and product identities. The integrand is 8 ^4 y ^2 y. We can express ^4 y ^2 y as ^2 y ( y y)^2. Using the identities: ^2 y = (1 - (2y))/(2) y y = ((2y))/(2) Substitute these into the expression: 8 ^4 y ^2 y = 8 ((1 - (2y))/(2)) (((2y))/(2))^2 = 8 ((1 - (2y))/(2)) ((^2(2y))/(4)) Now, apply the half-angle identity for ^2(2y): ^2(2y) = (1 - (2 · 2y))/(2) = (1 - (4y))/(2) Substitute this back: = 8 ((1 - (2y))/(2)) ((1 - (4y))/(2 · 4)) = 8