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Evaluate: $integral subscript 0 superscript straight pi over 4 end superscript space secx square root of fraction numerator 1 minus sin space straight x over denominator 1 plus sin space straight x end fraction end root dx.$

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112.

Evaluate: $integral subscript 0 superscript straight a space sin to the power of negative 1 end exponent space open parentheses square root of fraction numerator straight x over denominator straight a plus straight x end fraction end root close parentheses dx.$

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Short Answer Type

113.

Prove that:

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Long Answer Type

114.

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 4 cosx plus 2 sinx end fraction.$

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115.

Evaluate
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 2 space cosx space plus space 4 space sinx end fraction$

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116.

Evaluate
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx plus square root of 3 cosx end fraction dx$

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Short Answer Type

117.

Prove that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 9 plus 16 space cos squared straight x end fraction space equals space straight pi over 30$

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118.
Prove that:
$integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sin space 2 straight theta space dθ over denominator sin to the power of 4 space straight theta space plus space cos to the power of 4 straight theta end fraction space equals space straight pi over 4$

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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119.

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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120.

Prove that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cosx space dx over denominator left parenthesis 1 plus sin space straight x right parenthesis space left parenthesis 2 plus sinx right parenthesis end fraction space equals space log space 4 over 3$

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118. Prove that: