Question

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript 2 straight x space log space tanx space dx space equals space 0$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space log space tanx space dx$
$therefore space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript sin space 2 open parentheses straight pi over 2 minus straight x close parentheses. space log space tan open parentheses straight pi over 2 minus straight x close parentheses dx space space space rightwards double arrow space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space log space cot space straight x space dx$
$rightwards double arrow space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript space sin space 2 straight x space log open parentheses 1 over tanx close parentheses dx space space rightwards double arrow space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x left parenthesis log space 1 space minus space log space tanx right parenthesis space dx rightwards double arrow space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x left parenthesis 0 minus log space tanx right parenthesis space dx space rightwards double arrow space space straight I space equals space minus integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space log space tanx space dx$
$rightwards double arrow space space space straight I equals space minus straight I$ $open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
$rightwards double arrow space space space space 2 space straight I space equals space 0 space space space rightwards double arrow space space space straight I space equals space 0$

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Integrals

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript 2 straight x space log space tanx space dx space equals space 0$

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