Prove the inequality for "x>0" (otherwise it doesn’t hold)
"\\int\\limits_{0}^{x}{\\frac{1}{1+{{t}^{2}}}dt}={{\\tan }^{-1}}x"
By MVT
"\\int\\limits_{0}^{x}{\\frac{1}{1+{{t}^{2}}}dt}=\\frac{1}{1+{{\\xi }^{2}}}x,\\xi \\in \\left( 0,x \\right)"
We have
"\\frac{1}{1+{{\\xi }^{2}}}\\cdot x<1\\cdot x=x \\\\ \n\n\\frac{1}{1+{{\\xi }^{2}}}\\cdot x>\\frac{x}{1+{{x}^{2}}} \\\\"
Thus
"\\frac{x}{1+{{x}^{2}}}<{{\\tan }^{-1}}x<x"
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