Question

Show that:
$integral subscript 0 superscript infinity log space open parentheses straight x plus 1 over straight x close parentheses. space fraction numerator dx over denominator 1 plus straight x squared end fraction space equals space straight pi space log space 2$

Solution

Let I = $integral subscript 0 superscript infinity log space open parentheses straight x plus 1 over straight x close parentheses fraction numerator dx over denominator 1 plus straight x squared end fraction$
Put x = tan θ so that dx = sec² θ dθ
When $straight x equals 0 comma space space straight theta space equals space 0 comma space space space When space straight x rightwards arrow space infinity comma space space space straight theta rightwards arrow space straight pi over 2$
$therefore space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript log open parentheses tanθ plus 1 over tanθ close parentheses fraction numerator sec squared straight theta over denominator 1 plus tan squared straight theta end fraction dθ$

$equals space integral subscript 0 superscript straight pi over 2 end superscript log space open parentheses fraction numerator sin squared straight theta plus cos squared straight theta over denominator sinθ space cosθ end fraction close parentheses dθ space equals space integral subscript 0 superscript straight pi over 2 end superscript log space open parentheses fraction numerator 1 over denominator sin space straight theta space cosθ end fraction close parentheses dθ equals negative integral subscript 0 superscript straight pi over 2 end superscript log space sinθ space dθ space minus space integral subscript 0 superscript straight pi over 2 end superscript log space cos space straight theta space dθ space equals space minus open parentheses negative straight pi over 2 log space 2 close parentheses space minus open parentheses negative straight pi over 2 log space 2 close parentheses equals straight pi over 2 log space 2 space plus straight pi over 2 log space 2 space equals space straight pi space log space 2$

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Integrals

Show that:
$integral subscript 0 superscript infinity log space open parentheses straight x plus 1 over straight x close parentheses. space fraction numerator dx over denominator 1 plus straight x squared end fraction space equals space straight pi space log space 2$

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