Question

$integral subscript 0 superscript 2 over 3 end superscript fraction numerator dx over denominator 4 plus 9 straight x squared end fraction space equals$
$straight pi over 6$
$straight pi over 12$
$straight pi over 24$
$straight pi over 4$

Solution

straight pi over 24

Let
I = $integral subscript 0 superscript 2 over 3 end superscript fraction numerator dx over denominator 4 plus 9 straight x squared end fraction space equals space 1 over 9 integral subscript 0 superscript 2 over 3 end superscript fraction numerator 1 over denominator begin display style 4 over 9 end style plus straight x squared end fraction dx space equals space 1 over 9 integral subscript 0 superscript 2 over 3 end superscript fraction numerator 1 over denominator open parentheses begin display style 2 over 3 end style close parentheses squared plus straight x squared end fraction dx$
$equals space 1 over 9. fraction numerator 1 over denominator begin display style 2 over 3 end style end fraction open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator begin display style 2 over 3 end style end fraction close parentheses close square brackets subscript 0 superscript 2 over 3 end superscript space equals space 1 over 6 open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x over denominator 2 end fraction close parentheses close square brackets subscript 0 superscript 2 over 3 end superscript equals space 1 over 6 left parenthesis tan to the power of negative 1 end exponent space 1 space minus space tan to the power of negative 1 end exponent 0 right parenthesis space equals space 1 over 6 open parentheses straight pi over 4 minus 0 close parentheses space equals space straight pi over 24$

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Integrals

$integral subscript 0 superscript 2 over 3 end superscript fraction numerator dx over denominator 4 plus 9 straight x squared end fraction space equals$
$straight pi over 6$
$straight pi over 12$
$straight pi over 24$
$straight pi over 4$

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