Rewrite \({(x+1)}^{2}-1\) in the form \({a}^{2}-{b}^{2}\), where \(a=x+1\) and \(b=1\).
\[\imath ntegrate\sqrt{{(x+1)}^{2}-{1}^{2}}dxfrom\times 0to\times 1\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\imath ntegrate\sqrt{(x+1+1)(x+1-1)}dxfrom\times 0to\times 1\]
Simplify \(x+1+1\) to \(x+2\).
\[\imath ntegrate\sqrt{(x+2)(x+1-1)}dxfrom\times 0to\times 1\]
Simplify \(x+1-1\) to \(x\).
\[\imath ntegrate\sqrt{(x+2)x}dxfrom\times 0to\times 1\]
Regroup terms.
\[\imath ntegrate\sqrt{x(x+2)}dxfrom\times 0to\times 1\]
Simplify.
\[0\]
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