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111.
Evaluate: $integral subscript 0 superscript straight pi over 4 end superscript space secx square root of fraction numerator 1 minus sin space straight x over denominator 1 plus sin space straight x end fraction end root dx.$

straight I space equals space integral subscript 0 superscript straight pi over 4 end superscript secx square root of fraction numerator 1 minus sinx over denominator 1 plus sinx end fraction end root dx space equals space integral subscript 0 superscript straight pi over 4 end superscript secx square root of fraction numerator 1 minus sinx over denominator 1 plus sinx end fraction cross times fraction numerator 1 minus sinx over denominator 1 minus sinx end fraction end root dx

equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 1 over denominator cos space straight x end fraction. square root of fraction numerator left parenthesis 1 minus sinx right parenthesis squared over denominator 1 minus sin squared straight x end fraction end root dx space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 1 over denominator cos space straight x end fraction. square root of fraction numerator 1 minus sinx over denominator cos squared straight x end fraction end root dx

equals space integral subscript 0 superscript straight pi over 4 end superscript 1 over cosx. fraction numerator 1 minus sinx over denominator cosx end fraction dx space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 1 minus sinx over denominator cos squared straight x end fraction dx

equals space integral subscript 0 superscript straight pi over 4 end superscript space open parentheses fraction numerator 1 over denominator cos squared straight x end fraction minus 1 over cosx cross times sinx over cosx close parentheses dx space space equals integral subscript 0 superscript straight pi over 4 end superscript left parenthesis sec squared straight x minus secx space tanx right parenthesis space dx equals space open square brackets tanx space minus space secx close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space open parentheses tan straight pi over 4 minus sec straight pi over 4 close parentheses space minus space left square bracket tan space 0 space minus space sec space 0 right square bracket equals space open parentheses 1 minus square root of 2 close parentheses space minus space left parenthesis 0 space minus space 1 right parenthesis space equals space 1 minus square root of 2 space plus space 1 space equals space 2 minus square root of 2

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

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