Question

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator square root of cotx over denominator square root of tanx plus square root of cotx end fraction dx space equals space straight pi over 4$

Solution

Let I = $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator square root of cot space straight x end root over denominator square root of cot space straight x end root plus square root of tan space straight x end root end fraction dx$ ...(1)
Then I = $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator square root of cot open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end root over denominator square root of cot open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus square root of tan open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end root end root end fraction dx$ $open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator square root of tan space straight x end root over denominator square root of tan space straight x end root plus square root of cot space straight x end root end fraction dx$ ....(2)
Adding (1) and (2), we get,
$2 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript open square brackets fraction numerator square root of cot space straight x end root over denominator square root of cot space straight x end root plus square root of tan space straight x end root end fraction plus fraction numerator square root of tan space straight x end root over denominator square root of tan space straight x end root plus square root of cot straight x end fraction close square brackets dx$
$equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator square root of cot space straight x end root space plus space square root of tan space straight x over denominator square root of cot space straight x end root end fraction dx space equals space space integral subscript 0 superscript straight pi divided by 2 end superscript 1 space dx space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript space equals space straight pi over 2 minus 0 space equals space straight pi over 2$
$therefore space space space space 2 space straight I space equals space straight pi over 2 space space space space space space space rightwards double arrow space space space space space space space straight I space space space equals space straight pi over 4$

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Integrals

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator square root of cotx over denominator square root of tanx plus square root of cotx end fraction dx space equals space straight pi over 4$

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