fseries  =  Series[Exp[-x^2]*Sin[x]/x, {x, 0, 8}]

\(approxvalue\  = \ 
      N[\ ∫\_\(-π\)\%π Normal[fseries]\ \[DifferentialD]x] 
        \)

otherapprox  = 
      N[ (∫_(-π)^π) E^{-x^2} 
            Sin[x] / x \[DifferentialD]x]

secondapprox  =  
      N[ (∫_1^∞) Normal[fseries] \[DifferentialD]x]

secondotherapprox  = 
      N[ (∫_1^∞) E^{-x^2} 
            Sin[x] / x \[DifferentialD]x]

NIntegrate[Exp[-x^2] Sin[x] / x,  {x, 1, Infinity}]

yseries  =  Series[y[x], {x, 0, 10}]

Solve[
       {D[yseries, {x, 2}]  +  x * D[yseries, x]  +  yseries  ==  0, 
         y[0] == 1, y'[0] == 0}]

yseries