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^ 16 ^2 x ^2 x \,dx, we will use trigonometric identities to simplify the integrand. Step 1: Rewrite the integrand using the double angle identity for sine. We know that (2x) = 2 x x. Squaring both sides gives ^2(2x) = (2 x x)^2 = 4 ^2 x ^2 x. Therefore, ^2 x ^2 x = (1)/(4) ^2(2x). Substitute this into the integral: _0^ 16 ( (1)/(4) ^2(2x) ) \,dx _0^ 4 ^2(2x) \,dx Step 2: Use the power-reducing identity for ^2 . The identity is ^2 = (1 - (2))/(2). In our case, = 2x, so 2 = 4x. ^2(2x) = (1 - (4x))/(2) Substitute this into the integral: _0^ 4 ( (1 - (4x))/(2) ) \,dx _0^ 2 (1 - (4x)) \,dx _0^ (2 - 2(4x)) \,dx Step 3: Integrate the expression. (2 - 2(4x)) \,dx = 2x - 2 ( ((4x))/(4) ) + C = 2x - (1)/(2) (4x) + C Step 4: Evaluate the definite integral using the limits from 0 to . [ 2x - (1)/(2) (4x) ]_0^ = ( 2() - (1)/(2) (4) ) - ( 2(0) - (1)/(2) (4(0)) ) We know that (4) = 0 and (0) = 0. = ( 2 - (1)/(2)(0) ) - ( 0 - (1)/(2)(0) ) = 2 - 0 - 0 + 0 = 2 The final answer is 2.