Question

Prove that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 9 plus 16 space cos squared straight x end fraction space equals space straight pi over 30$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 9 plus 16 space cos squared straight x end fraction space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 9 space left parenthesis sin squared straight x plus cos squared straight x right parenthesis plus 16 cos squared straight x end fraction$
$equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 9 space sin squared straight x plus 25 space cos squared straight x end fraction space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator begin display style fraction numerator 1 over denominator cos squared space straight x end fraction end style dx over denominator begin display style fraction numerator 9 space sin squared straight x over denominator cos squared straight x end fraction end style plus begin display style fraction numerator 25 space cos squared straight x over denominator cos squared straight x end fraction end style end fraction$
   $equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sec squared xdx over denominator 9 space tan squared straight x plus 25 end fraction$
Put tan x = t,    ∴ sec² x dx = dt
When x = 0, t = tan 0 = 0
When    $straight x space equals space straight pi over 2 comma space space straight t space equals space tan straight pi over 2 space equals space infinity$
$therefore$    $straight I space equals space integral subscript 0 superscript infinity fraction numerator dt over denominator 9 straight t squared plus 25 end fraction space equals space 1 over 9 integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus begin display style 25 over 9 end style end fraction dt space equals space 1 over 9 integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus open parentheses begin display style 5 over 3 end style close parentheses squared end fraction dt$
$equals space 1 over 9. fraction numerator 1 over denominator begin display style 5 over 3 end style end fraction open square brackets tan to the power of negative 1 end exponent fraction numerator straight t over denominator open parentheses begin display style 5 over 3 end style close parentheses end fraction close square brackets subscript 0 superscript infinity space equals 1 over 15 open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight t over denominator 5 end fraction close parentheses close square brackets subscript 0 superscript infinity$
$equals space 1 over 15 open square brackets tan to the power of negative 1 end exponent infinity space minus space tan to the power of negative 1 end exponent 0 close square brackets space equals space 1 over 15 open square brackets straight pi over 2 minus 0 close square brackets space equals space straight pi over 30$

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Integrals

Prove that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 9 plus 16 space cos squared straight x end fraction space equals space straight pi over 30$

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