Find how many meters traveled during first t seconds

Question

A particle that moves along a straight line has velocity

v(t)=t^(2)e^(−4t)

meters per second after t seconds. How many meters will it travel during the first t seconds (from time=0 to time=t)?

Answer 1

The distance traveled is d(t) = ∫₀^t |v(s)| ds = ∫₀^t s²e^-s ; by integration by parts,

d(t) = s²(-e^-s) - (2s)(e^-s) + (2)(-e^-s)|₀^t = -t²e^-t -2te^-t -2e^-t - (-2) = 2 - (t² + 2t + 2)e^-t