Question

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator straight theta over denominator sin space straight theta end fraction close parentheses squared space dθ space equals space straight pi space log space 2$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator straight theta over denominator sin space straight theta end fraction close parentheses squared space dθ space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator straight theta squared over denominator sin squared space straight theta end fraction dθ space equals space integral subscript 0 superscript straight pi over 2 end superscript straight theta squared space cosec squared straight theta space dθ$
$equals space open square brackets straight theta squared left parenthesis negative cot space straight theta right parenthesis close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space integral subscript 0 superscript straight pi over 2 end superscript 2 space straight theta. space left parenthesis negative cot space straight theta right parenthesis space dθ space equals space open square brackets straight theta squared space left parenthesis negative cot space straight theta right parenthesis close square brackets subscript 0 superscript straight pi over 2 end superscript plus 2 integral subscript 0 superscript straight pi over 2 end superscript straight theta. space cot space straight theta space dθ space equals space open parentheses negative straight pi squared over 4 cot straight pi over 2 plus 0 close parentheses plus 2 open curly brackets open square brackets straight theta space. log space sinθ close square brackets subscript 0 superscript straight pi over 2 end superscript minus integral subscript 0 superscript straight pi over 2 end superscript 1. space log space sinθ space dθ close curly brackets equals space straight pi squared over 4 cross times 0 plus 0 plus 2 open square brackets straight theta. space log space sinθ close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space 2 integral subscript 0 superscript straight pi over 2 end superscript log space sinθ space dθ space equals 2 open square brackets straight pi over 2 log space sin straight pi over 2 minus 0 close square brackets space minus space 2 open parentheses negative straight pi over 2 log space 2 close parentheses space equals space 2 space left square bracket 0 minus 0 right square bracket space plus space straight pi space log space 2 space space equals space straight pi space log space 2$

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Integrals

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator straight theta over denominator sin space straight theta end fraction close parentheses squared space dθ space equals space straight pi space log space 2$

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