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

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 4 space cosx plus 2 sinx end fraction dx$
Put $tan straight x over 2 space equals space straight t comma space space space space space therefore space space space straight x over 2 space equals space tan to the power of negative 1 end exponent space straight t space space rightwards double arrow space space space straight x space equals space 2 space tan to the power of negative 1 end exponent straight t space space space space rightwards double arrow space space space dx space equals space fraction numerator 2 over denominator 1 plus straight t squared end fraction dt$
When x = 0, $straight t equals space tan 0 space equals space 0$

When $straight x equals space straight pi over 2 comma space space space space space space straight t space equals space tan straight pi over 4 equals 1$
Also    $sinx space equals space fraction numerator 2 tan begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals space fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction. space cos space straight x space equals space fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals space fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction$

   $straight I space equals space integral subscript 0 superscript 1 fraction numerator begin display style fraction numerator 2 over denominator 1 plus straight t squared end fraction end style dt over denominator 4 open parentheses begin display style fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end style close parentheses plus 2 open parentheses begin display style fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end style close parentheses end fraction space equals integral subscript 0 superscript 1 fraction numerator 1 over denominator 2 space left parenthesis 1 minus straight t squared right parenthesis space plus space 2 straight t end fraction dt$

   $equals 1 half integral subscript 0 superscript 1 fraction numerator 1 over denominator 1 plus straight t minus straight t squared end fraction space equals space 1 half integral fraction numerator 1 over denominator 1 minus left parenthesis straight t squared minus straight t right parenthesis end fraction dt space equals space 1 half integral fraction numerator 1 over denominator begin display style 5 over 4 end style minus open parentheses straight t squared minus straight t plus begin display style 1 fourth end style close parentheses end fraction dt$

$equals space 1 half integral fraction numerator 1 over denominator open parentheses begin display style fraction numerator square root of 5 over denominator 2 end fraction end style close parentheses squared minus open parentheses straight t minus begin display style 1 over straight t end style close parentheses squared end fraction dt space equals space 1 half space equals space fraction numerator 1 over denominator 2 begin display style fraction numerator square root of 5 over denominator 2 end fraction end style end fraction open square brackets log open vertical bar fraction numerator begin display style fraction numerator square root of 5 over denominator 2 end fraction end style plus open parentheses straight t minus begin display style 1 half end style close parentheses over denominator begin display style fraction numerator square root of 5 over denominator 2 end fraction end style minus open parentheses straight t minus begin display style 1 half end style close parentheses end fraction close vertical bar close square brackets subscript 0 superscript 1$

$equals fraction numerator 1 over denominator 2 square root of 5 end fraction open square brackets log space open parentheses fraction numerator square root of 5 plus 1 over denominator square root of 5 minus 1 end fraction close parentheses minus log open parentheses fraction numerator square root of 5 minus 1 over denominator square root of 5 plus 1 end fraction close parentheses close square brackets equals space fraction numerator 1 over denominator 2 square root of 5 end fraction log open parentheses fraction numerator square root of 5 plus 1 over denominator square root of 5 minus 1 end fraction cross times fraction numerator square root of 5 plus 1 over denominator square root of 5 minus 1 end fraction close parentheses space equals space fraction numerator 1 over denominator 2 square root of 5 end fraction log open parentheses fraction numerator 5 plus 1 plus 2 square root of 5 over denominator 5 plus 1 minus 2 square root of 5 end fraction close parentheses equals space fraction numerator 1 over denominator 2 square root of 5 end fraction log open parentheses fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction close parentheses space equals space fraction numerator 1 over denominator 2 square root of 5 end fraction log open parentheses fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction cross times fraction numerator 3 plus square root of 5 over denominator 3 plus square root of 5 end fraction close parentheses equals space fraction numerator 1 over denominator 2 square root of 5 end fraction log open square brackets fraction numerator left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 4 end fraction close square brackets space equals space fraction numerator 1 over denominator 2 square root of 5 end fraction log open parentheses fraction numerator 3 plus square root of 5 over denominator 2 end fraction close parentheses squared space equals space fraction numerator 1 over denominator square root of 5 end fraction log open parentheses fraction numerator 3 plus square root of 5 over denominator 2 end fraction close parentheses$

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

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