Question

Prove that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space 2 straight ϕ space dϕ over denominator sin to the power of 4 straight ϕ plus cos to the power of 4 straight ϕ end fraction space equals space straight pi over 2$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space 2 straight ϕ over denominator sin to the power of 4 straight ϕ plus cos to the power of 4 straight ϕ end fraction dϕ space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 2 sinϕcosϕ over denominator sin to the power of 4 straight ϕ plus cos to the power of 4 straight ϕ end fraction dϕ$
$equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator begin display style fraction numerator 2 space sin space straight ϕ space cosϕ over denominator cos to the power of 4 straight ϕ end fraction end style over denominator begin display style fraction numerator sin to the power of 4 straight ϕ over denominator cos to the power of 4 straight ϕ end fraction end style plus begin display style fraction numerator cos to the power of 4 straight ϕ over denominator cos to the power of 4 straight ϕ end fraction end style end fraction dϕ space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 2 space tanϕ space sec squared space straight ϕ space dϕ over denominator tan to the power of 4 straight ϕ plus 1 end fraction$
Put tan² ϕ = y , ∴ 2 tan ϕ sec² ϕ dϕ = dy When ϕ = 0, y = tan² 0 = 0
When $straight ϕ space equals space straight pi over 2 comma space space straight y space equals tan squared straight pi over 2 space rightwards arrow space space infinity$
$therefore space space space space space straight I space equals space integral subscript 0 superscript infinity fraction numerator dy over denominator straight y squared plus 1 end fraction space equals space open square brackets tan to the power of negative 1 end exponent space straight y close square brackets subscript 0 superscript infinity space equals space tan to the power of negative 1 end exponent infinity space minus space tan to the power of negative 1 end exponent 0 space equals space straight pi over 2 minus 0 space equals space straight pi over 2$

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Integrals

Prove that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space 2 straight ϕ space dϕ over denominator sin to the power of 4 straight ϕ plus cos to the power of 4 straight ϕ end fraction space equals space straight pi over 2$

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