Question

Evaluate $integral subscript 0 superscript straight pi over 4 end superscript space log space left parenthesis 1 plus tanx right parenthesis space dx.$

Solution

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space log left parenthesis 1 plus tanx right parenthesis space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript log open square brackets 1 plus tan open parentheses straight pi over 4 minus straight x close parentheses close square brackets dx$
   $open square brackets because space space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space log open square brackets 1 plus fraction numerator 1 minus tanx over denominator 1 plus tanx end fraction close square brackets dx space equals space integral subscript 0 superscript straight pi over 4 end superscript space log space open square brackets fraction numerator 1 plus tanx plus 1 minus tanx over denominator 1 plus tanx end fraction close square brackets dx$
   $equals space integral subscript 0 superscript straight pi over 4 end superscript log open parentheses fraction numerator 2 over denominator 1 plus tanx end fraction close parentheses dx space equals space integral subscript 0 superscript straight pi over 4 end superscript log space 2 space dx space minus space integral subscript 0 superscript straight pi over 4 end superscript log left parenthesis 1 plus tanx right parenthesis space dx$
$therefore space space space straight I space equals space log space 2 integral subscript 0 superscript straight pi over 4 end superscript space 1 space dx space minus space 1$ $open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
$rightwards double arrow space space space 2 space straight I space equals space log space 2 space open square brackets straight x close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space log space 2 open square brackets straight pi over 4 minus 0 close square brackets space space space space space space rightwards double arrow space space space 2 space straight I space equals space straight pi over 4 log 2 space rightwards double arrow space straight I space equals space straight pi over 8 log space 2$

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Integrals

Evaluate $integral subscript 0 superscript straight pi over 4 end superscript space log space left parenthesis 1 plus tanx right parenthesis space dx.$

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