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

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx plus square root of 3 cosx end fraction dx$
Put    $1 space equals straight r space cos space straight alpha$ ...(1)
and    $square root of 3 space equals space straight r space sin space straight alpha$ ...(2)
Squaring and adding (1) and (2), we get
$4 space equals space straight r squared$ $rightwards double arrow space space space straight r space equals space 2$
Dividing (2) by (1), $tan space straight alpha space equals space square root of 3$ $rightwards double arrow space space space straight a space equals space straight pi over 3$
$therefore$ $straight I space equals space 1 over straight r integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx space cosα space plus space cosx space sinα end fraction dx space equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sin left parenthesis straight x plus straight alpha right parenthesis end fraction dx$
   $equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript cosec left parenthesis straight x plus straight alpha right parenthesis space dx space equals space 1 half open square brackets negative log space open vertical bar cosec left parenthesis straight x plus straight alpha right parenthesis plus cot left parenthesis straight x plus straight a right parenthesis close vertical bar close square brackets subscript 0 superscript straight pi over 2 end superscript$

$equals space minus 1 half open square brackets log space open vertical bar cosec space open parentheses straight pi over 2 plus straight pi over 3 close parentheses plus cot space open parentheses straight pi over 2 plus straight pi over 3 close parentheses close vertical bar minus log open vertical bar cosec straight pi over 3 plus cot straight pi over 3 close vertical bar close square brackets$

$equals negative 1 half open square brackets log space open vertical bar sec straight pi over 3 minus tan straight pi over 3 close vertical bar minus log open vertical bar cosec straight pi over 3 plus cot straight pi over 3 close vertical bar close square brackets$

   $equals negative 1 half open square brackets log space open vertical bar 2 minus square root of 3 close vertical bar minus log space open vertical bar fraction numerator 2 over denominator square root of 3 end fraction plus fraction numerator 1 over denominator square root of 3 end fraction close vertical bar close square brackets equals negative 1 half open square brackets log left parenthesis 2 minus square root of 3 right parenthesis space minus space log square root of 3 close square brackets equals space minus 1 half log open parentheses fraction numerator 2 minus square root of 3 over denominator square root of 3 end fraction close parentheses$

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

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