Prove the following:
$integral subscript 0 superscript straight pi over 4 end superscript space 2 space tan cubed straight x space dx space equals space 1 minus log space 2.$

Solution

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space 2 space tan cubed straight x space dx space equals space 2 integral subscript 0 superscript straight pi over 4 end superscript tanx. space tan squared straight x space dx space equals space 2 space integral subscript 0 superscript straight pi over 4 end superscript tan space straight x left parenthesis sec squared straight x minus 1 right parenthesis space dx$
$equals space 2 space integral subscript 0 superscript straight pi over 4 end superscript space tanx space sec squared straight x space dx space minus space 2 integral subscript 0 superscript straight pi over 4 end superscript space tanx space dx space equals space 2 open square brackets fraction numerator tan squared straight x over denominator 2 end fraction close square brackets subscript 0 superscript straight pi over 4 end superscript plus 2 open square brackets log open vertical bar cos space straight x close vertical bar close square brackets subscript 0 superscript straight pi over 4 end superscript$
$equals space open parentheses tan squared straight pi over 4 minus tan squared 0 close parentheses space plus space 2 open square brackets log space open vertical bar cos space straight pi over 4 close vertical bar minus log open vertical bar cos space 0 close vertical bar close square brackets$
$equals space open vertical bar 1 minus 0 close vertical bar space plus space 2 open parentheses log space fraction numerator 1 over denominator square root of 2 end fraction minus log space 1 close parentheses space equals space 1 plus 2 space left parenthesis log space 1 space minus space log space square root of 2 minus log space 1 right parenthesis equals space 1 minus 2 space log space square root of 2 space equals space 1 minus log left parenthesis square root of 2 right parenthesis squared space equals space 1 minus log space 2.$

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Integrals

Prove the following:
$integral subscript 0 superscript straight pi over 4 end superscript space 2 space tan cubed straight x space dx space equals space 1 minus log space 2.$

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