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S ( x e^x ) dx
= S ( x ) d (e^x)
= x e^x - S (e^x) dx
= x e^x - e^x + C

S [ e^( x^2 ) ] dx
This integration known be have no analytical result (unless using multivariable calculus only on special cases). Do not try it on or below A-level math.

x e^x - e^x + C

S [ e^( x^2 ) ] dx