<< graphics`graphics`

task 1

a

Integrate[
      E^(-a t) E^(-2 a τ), {τ, 0, ∞}, 
      Assumptions -> Re[a] > 0]

Plot[((E^(-10 t))/(2 10)), {t, -1, 
        5}]

Plot[((E^(-0.1 t))/(2 0.1)), {t, -5, 
        5}]

b

Integrate[(UnitStep[τ] - 
            UnitStep[τ - 1]) (UnitStep[τ + t] - 
            UnitStep[τ + t - 1]), {τ, 0, ∞}]

UnitStep[τ] - UnitStep[τ - 1]

Plot[-(1 + t) UnitStep[-1 - t] - (-1 + 
              t) UnitStep[1 - t] + 2 t UnitStep[-t], {t, -10, 
        10}]

task 2

a

Integrate[Cos[x a]/(1 + x^2), {x, -∞, ∞}, 
      Assumptions -> {Im[a] == 0, a > 0}]

Integrate[
      Sin[x t]/(1 + x^2), {x, -∞, ∞}]

Integrate[
      Cos[x t] E^(-a x^2), {x, -∞, ∞}, 
      Assumptions -> {t ∈ Reals && 
            Re[a] ≥ 0}]

Integrate[
      Sin[x t] E^(-a x^2), {x, -∞, ∞}]

Plot[Cos[16 t] (E^(-t)) , {t, 0, 2}, 
      ImageSize -> 550]

DisplayTogether[Plot[Cos[16 t] (E^(-t)) , {t, 0, 2}], 
      Plot[(E^(-t)) , {t, 0, 2}], 
      ImageSize -> 550]

DisplayTogether[Plot[Cos[16 t] Exp[-t^2], {t, 0, 2}], 
      Plot[Exp[-t^2], {t, 0, 2}], ImageSize -> 550]

Plot[Cos[16 t] Exp[-t^2], {t, 0, 2}, 
      ImageSize -> 550]

DisplayTogether[
      Plot[Cos[t] (E^(-Abs[t])) , {t, 0, 5}, 
        ImageSize -> 550], 
      Plot[Cos[t] Exp[-t^2], {t, 0, 5}, 
        ImageSize -> 550]]