Question

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus cot cubed straight x end fraction dx$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus cot cubed straight x end fraction dx space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus begin display style fraction numerator cos cubed straight x over denominator sin cubed straight x end fraction end style end fraction dx$
$therefore space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin cubed straight x over denominator sin cubed straight x plus cos cubed straight x end fraction dx$ ...(1)
$rightwards double arrow space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin cubed open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin cubed open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos cubed open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx$
$therefore space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos cubed straight x over denominator cos cubed straight x plus sin cubed straight x end fraction dx$ ...(2)
Adding (1) and (2), we get.
$2 space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin cubed straight x plus cos cubed straight x over denominator sin cubed straight x plus cos cubed straight x end fraction dx$
$therefore$ $2 space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript 1 space dx space space space rightwards double arrow space space space space space 2 space straight I space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi over 2 end superscript$
$therefore space space space space space 2 space straight I space equals space straight pi over 2 minus 0 space space space rightwards double arrow space space straight I space equals space straight pi over 4$

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Integrals

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus cot cubed straight x end fraction dx$

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