Is kinetic energy the same as the distance traveled?

Because if you were to derive 1/2v^2 its derivative would be velocity with respect to time. If you were to integrate velocity with respect to time, wouldn't you find the distance traveled? and since mass is constant (for the most part) doesn't that equal 1/2mv^2. Can you tell me my error (if there's one) and tell me how they solved the formula for KE