Question

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript space square root of sin space straight ϕ end root space cos to the power of 5 straight ϕ space dϕ space$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript square root of sin space straight ϕ end root space cos to the power of 5 straight ϕ space dϕ space equals space integral subscript 0 superscript straight pi over 2 end superscript square root of sin space straight ϕ end root. space cos to the power of 4 straight ϕ. space cosϕ space dϕ$
   $equals space integral subscript 0 superscript straight pi over 2 end superscript square root of sin space straight ϕ end root. space left parenthesis cos squared straight ϕ right parenthesis squared space cos space straight ϕ space dϕ space equals space integral subscript 0 superscript straight pi over 2 end superscript square root of sin space straight ϕ end root space left parenthesis 1 minus sin squared straight ϕ right parenthesis squared. space cos space straight ϕ space dϕ$
Put sin ϕ = t,    ∴ cos ϕ dϕ = dt
When ϕ = 0, t = sin 0 = 0
When $straight ϕ space equals space straight pi over 2 comma space straight t space equals space sin space straight pi over 2 space equals space 1$
$therefore$    $straight I space equals space integral subscript 0 superscript 1 square root of straight t left parenthesis 1 minus straight t right parenthesis squared space dt space equals space integral subscript 0 superscript 1 straight t to the power of 1 half end exponent left parenthesis straight t to the power of 4 minus 2 straight t squared plus 1 right parenthesis space dt$
$equals space integral subscript 0 superscript 1 open parentheses straight t to the power of 9 over 2 end exponent minus 2 straight t to the power of 5 over 2 end exponent plus straight t to the power of 1 half end exponent close parentheses space dt space equals space open square brackets fraction numerator straight t to the power of begin display style 11 over 2 end style end exponent over denominator begin display style 11 over 2 end style end fraction minus 2 fraction numerator straight t to the power of begin display style 7 over 2 end style end exponent over denominator begin display style 7 over 2 end style end fraction plus fraction numerator straight t to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 1$
$open square brackets 2 over 11 straight t to the power of 11 over 2 end exponent minus 4 over 7 straight t to the power of 7 over 2 end exponent plus 2 over 3 straight t to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 1 space equals space open square brackets open parentheses 2 over 11 minus 4 over 7 plus 2 over 3 close parentheses minus left parenthesis 0 plus 0 plus 0 right parenthesis close square brackets$
$equals space fraction numerator 42 minus 132 plus 154 over denominator 231 end fraction equals 64 over 231$

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Integrals

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript space square root of sin space straight ϕ end root space cos to the power of 5 straight ϕ space dϕ space$

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