`\!\(
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RowBox[{
SubsuperscriptBox["\[Integral]", "1", "2"],
RowBox[{
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SuperscriptBox["x", "2"], "+", "x", "+", "1"}],
SuperscriptBox["x", "2"]],
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HoldForm]\)`
`%//N`
Gives me $2.19315$, which is certainly an approximation.