Math. 487 In-Class Notebook:
      Thursday, January 22, 2004

∫E^(-x^2) (Sin[x])^2/
          x \[DifferentialD]x

(∫_(-π)^π) E^(-x^2) (Sin[
                x])^2/x \[DifferentialD]x

eseries  =  Series[E^x, {x, 0, 3}]

SpecialSeries  =  (eseries - 1)/x

SpecialSeries2  =  
      Series[(E^x - 1)/x, {x, 0, 2}]

SpecialSeries  -  SpecialSeries2

eminusxsq  =  
      Series[E^(-x^2), {x, 0, 6}]

sine  =  Series[Sin[x], {x, 0, 6}]

sinesq  =  sine^2

numerator  =  eminusxsq * sinesq

integrand  =  numerator / x

f[x_]  =  Normal[integrand]

f[1]

N[f[1]]

(∫_(-π)^π) f[x] \[DifferentialD]x

Some considerations about ODE's

DSolve[y'[t] == y[t], y[t], t]

ODEsolution  = 
      DSolve[{y'[t] == y[t], y[0] == 1}, y[t], t]

g[t_]  =  y[t]  /.  First[ODEsolution]

N[g[1], 20]

Plot[g[t], {t, -1, 1}]

g2[t_]  =  Normal[Series[g[t], {t, 0, 2}]]

Plot[{g[t], g2[t]}, {t, -1, 1},  
      PlotStyle ->  {Hue[0.5], Hue[0.9]}]