Question

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

Solution

Let I = $integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sin space 2 straight theta over denominator sin to the power of 4 straight theta space plus space cos to the power of 4 straight theta end fraction dθ space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 2 space sin space straight theta space cos space straight theta space dθ over denominator sin to the power of 4 straight theta space plus space cos to the power of 4 straight theta end fraction$
$equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator begin display style fraction numerator 2 space sin space straight theta space cosθ over denominator cos to the power of 4 straight theta end fraction end style over denominator begin display style fraction numerator sin to the power of 4 straight theta over denominator cos to the power of 4 straight theta end fraction end style plus begin display style fraction numerator cos to the power of 4 straight theta over denominator cos to the power of 4 straight theta end fraction end style end fraction dθ space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 2 space tanθ space sec squared straight theta over denominator tan to the power of 4 straight theta plus 1 end fraction dθ$
Put tan² θ = t, ∴ 2 tan θ sec² θ dθ = dt When θ = 0, t = tan² 0 = 0
When $straight theta space equals space straight pi over 4 comma space space space straight t space space equals tan squared straight pi over 4 space equals space 1$
$therefore$ $straight I space equals space integral subscript 0 superscript 1 fraction numerator dt over denominator straight t squared plus 1 end fraction space equals space left square bracket tan to the power of negative 1 end exponent straight t right square bracket subscript 0 superscript 1 space equals space tan to the power of negative 1 end exponent 1 space minus space tan to the power of negative 1 end exponent 0 space equals space straight pi over 4 minus 0 space equals space straight pi over 4$

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