Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space tan to the power of negative 1 end exponent left parenthesis sin space straight x right parenthesis space dx.$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript sin space 2 space straight x space tan to the power of negative 1 end exponent space left parenthesis sin space straight x right parenthesis space dx space equals space integral subscript 0 superscript straight pi over 2 end superscript space 2 sinx space cosx. space tan to the power of negative 1 end exponent left parenthesis sinx right parenthesis space dx$
Put sin x = t,    ∴ cos x dx = dt
When x = 0. t = sin 0 = 0
When $straight x space equals space straight pi over 2. space space space space space straight t space equals space sin straight pi over 2 space equals space 1$
$therefore$ $straight I space equals space 2 integral subscript 0 superscript 1 straight t. space tan to the power of negative 1 end exponent straight t space dt space equals space 2 integral subscript 0 superscript 1 left parenthesis tan to the power of negative 1 end exponent straight t right parenthesis space straight t space dt$
   $equals 2 open square brackets tan to the power of negative 1 end exponent straight t. straight t squared over 2 close square brackets subscript 0 superscript 1 space minus space 2 integral subscript 0 superscript 1 fraction numerator 1 over denominator 1 plus straight t squared end fraction. straight t squared over 2 dt$
   $equals space open square brackets straight t squared space tan to the power of negative 1 end exponent space straight t close square brackets subscript 0 superscript 1 space minus space integral subscript 0 superscript 1 space fraction numerator straight t squared over denominator 1 plus straight t squared end fraction dt space space equals space left parenthesis tan to the power of negative 1 end exponent 1 minus 0 right parenthesis minus integral subscript 0 superscript 1 open parentheses 1 minus fraction numerator 1 over denominator 1 plus straight t squared end fraction close parentheses dt$
   $equals space straight pi over 4 minus open square brackets straight t minus tan to the power of negative 1 end exponent straight t close square brackets subscript 0 superscript 1 space equals space straight pi over 4 minus left square bracket left parenthesis 1 minus tan to the power of negative 1 end exponent right parenthesis space minus space left parenthesis 0 minus tan to the power of negative 1 end exponent 0 right parenthesis right square bracket equals space straight pi over 4 minus open square brackets open parentheses 1 minus straight pi over 4 close parentheses minus left parenthesis 0 minus 0 right parenthesis close square brackets space equals space straight pi over 2 minus 1$

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Integrals

Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space tan to the power of negative 1 end exponent left parenthesis sin space straight x right parenthesis space dx.$

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