Final answer:
To find the equation for x(t), match the derivative of the position function with the given velocity and use the initial condition x(1) = 0 to find constant C. The final position function is x(t) = tlnt - 3t + 3.
Step-by-step explanation:
The question asks us to find the equation for x(t), which is the position of a particle moving along the x-axis, given that the initial condition is x(1) = 0 (particle is at the origin when t = 1) and the velocity is given by an integral v(t) = ∠t - 2. Since we have the integral form of the velocity, we can differentiate the position function x(t) to find the velocity function, and then equate it to the given velocity integral to find the constant C. After differentiating x(t) = tlnt - 3t + C, we get v(t) = lnt + 1 - 3. Setting this equal to the given v(t), we match the integrand with the derivative and get lnt + 1 - 3 = t - 2, which simplifies to lnt - 2 = t - 3. Since x(1) = 0, we substitute t = 1 into the original x(t) equation to find C. The equation simplifies to 0 = 1·ln(1) - 3·1 + C, which results in C = 3 because ln(1) = 0. Plugging C back into the original equation, we get the final position function as x(t) = tlnt - 3t + 3.