Final answer:
To evaluate the integral, we can use a substitution technique. After substituting and simplifying, the integral becomes -1/24(1 - x⁴)^(3/2). Evaluating this from 0 to 1/√2 gives the final answer of -1/24(1/2)^(3/2) + 1/24.
Step-by-step explanation:
To evaluate the integral ∫(x * √(1 - x⁴)) dx from 0 to 1/√2, we can use a substitution. Let u = 1 - x⁴. Then, the differential du = -4x³dx. Rearranging, we have dx = -du/(4x³). Substituting these values into the integral, we get:
∫(x * √(1 - x⁴)) dx = ∫(-x * √u/(4x³)) (-du/(4x³)) = -∫(√u du/(16x⁴)).
We can rewrite the integral as -1/16∫u^0.5 du, which can be integrated as -1/16 * (2/3) * u^(3/2) + C = -1/24u^(3/2) + C.
Substituting back in x, the final answer is -1/24(1 - x⁴)^(3/2) + C. Evaluating this from 0 to 1/√2:
-1/24(1 - (1/√2)⁴)^(3/2) - (-1/24(1 - 0⁴)^(3/2) = -1/24(1 - 1/2)^(3/2) + 1/24(1 - 0)^(3/2) = -1/24(1/2)^(3/2) + 1/24.