$integral subscript 1 superscript square root of 3 end superscript fraction numerator dx over denominator 1 plus straight x squared end fraction$ equals

$straight pi over 3$
$fraction numerator 2 straight pi over denominator 3 end fraction$
$straight pi over 6$
$straight pi over 12$

Solution

straight pi over 12

Let I = $integral subscript 1 superscript square root of 3 end superscript fraction numerator 1 over denominator 1 plus straight x squared end fraction dx space equals space open square brackets tan to the power of negative 1 end exponent straight x close square brackets subscript 1 superscript square root of 3 end superscript$
$equals space tan to the power of negative 1 end exponent square root of 3 space minus space tan to the power of negative 1 end exponent 1 space equals space straight pi over 3 minus straight pi over 4 space equals space fraction numerator 4 straight pi minus 3 straight pi over denominator 12 end fraction equals straight pi over 12$

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Integrals

$integral subscript 1 superscript square root of 3 end superscript fraction numerator dx over denominator 1 plus straight x squared end fraction$ equals

$straight pi over 3$
$fraction numerator 2 straight pi over denominator 3 end fraction$
$straight pi over 6$
$straight pi over 12$

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