Prove the following:
$integral subscript 0 superscript 1 sin to the power of negative 1 end exponent straight x space dx space equals space straight pi over 2 minus 1$

Solution

Let I = $integral subscript 0 superscript 1 sin to the power of negative 1 end exponent straight x space dx space equals space integral subscript 0 superscript 1 sin to the power of negative 1 end exponent straight x.1 dx space equals space left square bracket sin to the power of negative 1 end exponent straight x. space straight x right square bracket subscript 0 superscript 1 space minus space integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction. straight x space dx$
$equals space left square bracket space straight x space sin to the power of negative 1 end exponent space straight x right square bracket subscript 0 superscript 1 space plus space 1 half integral subscript 0 superscript 1 left parenthesis 1 minus straight x squared right parenthesis to the power of negative 1 half end exponent left parenthesis negative 2 straight x right parenthesis space dx$
$equals space open square brackets straight x space sin to the power of negative 1 end exponent space straight x close square brackets subscript 0 superscript 1 space plus 1 half open square brackets fraction numerator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction close square brackets subscript 0 superscript 1 space equals space left square bracket straight x space sin to the power of negative 1 end exponent space straight x right square bracket subscript 0 superscript 1 space plus space open square brackets square root of 1 minus straight x squared end root close square brackets subscript 0 superscript 1$
$equals left parenthesis sin to the power of negative 1 end exponent 1 minus 0 right parenthesis space plus space left square bracket 0 minus 1 right square bracket space equals space straight pi over 2 minus 1$

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Integrals

Prove the following:
$integral subscript 0 superscript 1 sin to the power of negative 1 end exponent straight x space dx space equals space straight pi over 2 minus 1$

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