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I am not sure how to put this in good format, but I want to plot this on Mathematica:

$$ u(t) = \frac{1}{4\pi t} \int\mathrm dy\ \mathrm e^{-y^2/(4t)}\ y\ (1 - y^2) $$

on the interval $ [a, b] $ where $ a = 0 $ and $ b = 1 $ are the upper and lower bounds.

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The integral given in the question can be done symbolically.

Clear["Global`*"]

u[t_] = Integrate[1/(4*Pi*t)*E^(-y^2/(4*t))*y (1 - y^2), {y, 0, 1}] // 
  FullSimplify

(* (1 + 4 (-1 + E^(-(1/4)/t)) t)/(2 π) *)

EDIT: As suggested by Artes

Limit[u[t], t -> 0, Direction -> #] & /@
 {"FromAbove", "FromBelow"}

(* {1/(2 π), -∞} *)

Plot[u[t], {t, -5, 5}, PlotRange -> {-0.06, 0.175}]

enter image description here

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