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Integrals

Short Answer Type

121.

Evaluate: $integral subscript 0 superscript straight pi over 4 end superscript space fraction numerator sinx space cosx over denominator cos squared straight x plus sin to the power of 4 straight x end fraction dx.$

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122.

Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sinx space cosx over denominator cos to the power of 4 straight x space plus space sin to the power of 4 straight x end fraction dx$

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123.

Evaluate the following definite integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of sin space 2 straight x end root end fraction dx.$

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124.

Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sinx plus cosx over denominator 9 plus 16 sin 2 straight x end fraction dx$

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125.

Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space tan to the power of negative 1 end exponent left parenthesis sin space straight x right parenthesis space dx.$

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126.
Prove that $integral subscript 0 superscript straight pi fraction numerator dx over denominator 5 plus 3 cosx end fraction space equals space straight pi over 4$

Let
I = $integral subscript 0 superscript straight pi fraction numerator dx over denominator 5 plus 3 space cosx end fraction$

Put $tan straight x over 2 space equals space straight t$ or    $straight x over 2 space equals space tan to the power of negative 1 end exponent straight t$ or    $straight x equals 2 tan to the power of negative 1 end exponent straight t space space space space space space rightwards double arrow space space space dx space equals space fraction numerator 2 over denominator 1 plus straight t squared end fraction dt$
Also    $cosx space equals space fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals space fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction$

When    $straight x space equals space 0 comma space space space space space space space space straight t space equals space tan space 0 space equals space 0$
When    $straight x equals space straight pi comma space space space space space straight t space equals space tan straight pi over 2 space equals space infinity$
$therefore$ I = $integral subscript 0 superscript infinity fraction numerator begin display style fraction numerator 2 over denominator 1 plus straight t squared end fraction end style dt over denominator 5 plus 3 open parentheses begin display style fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end style close parentheses end fraction space equals space integral subscript 0 superscript infinity fraction numerator 2 over denominator 5 space left parenthesis 1 plus straight t squared right parenthesis plus 3 space left parenthesis 1 minus straight t squared right parenthesis end fraction dt$
   $equals space integral subscript 0 superscript infinity fraction numerator 2 over denominator 2 straight t squared plus 8 end fraction dt space equals space integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus 4 end fraction dt space equals space integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus left parenthesis 2 right parenthesis squared end fraction dt equals space 1 half. space space space open square brackets tan to the power of negative 1 end exponent straight t over 2 close square brackets subscript 0 superscript infinity space equals space 1 half left parenthesis tan to the power of negative 1 end exponent infinity space minus tan to the power of negative 1 end exponent 0 right parenthesis space equals space 1 half open parentheses straight pi over 2 minus 0 close parentheses space equals straight pi over 4.$

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127.

Evaluate $integral subscript 0 superscript straight pi fraction numerator dx over denominator 5 plus 4 space cosx end fraction.$

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128.

Evaluate the following:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 5 plus 4 space sinx end fraction$