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

Let
I = $integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sinx space cosx over denominator cos to the power of 4 straight x plus sin to the power of 4 straight x end fraction dx space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator begin display style fraction numerator sinx space cosx over denominator cos to the power of 4 straight x end fraction end style over denominator begin display style fraction numerator cos to the power of 4 straight x over denominator cos to the power of 4 straight x end fraction end style plus begin display style fraction numerator sin to the power of 4 straight x over denominator cos to the power of 4 straight x end fraction end style end fraction dx$
$equals integral subscript 0 superscript straight pi over 4 end superscript fraction numerator begin display style sinx over cosx end style minus begin display style fraction numerator 1 over denominator cos squared straight x end fraction end style over denominator 1 plus begin display style fraction numerator sin to the power of 4 straight x over denominator cos to the power of 4 straight x end fraction end style end fraction dx space equals integral subscript 0 superscript straight pi over 4 end superscript fraction numerator tan space straight x space sec squared straight x over denominator 1 plus tan to the power of 4 straight x end fraction dx$
Put tan ² x = y, ∴ 2 tan x sec² x dx = dy

$therefore$ When x = 0, y = tan² 0 = 0
When $straight x equals space straight pi over 4 comma space space space straight y space equals space tan squared straight pi over 4 equals space 1$
$therefore space space space space straight I space equals space 1 half integral subscript 0 superscript 1 fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space 1 half open square brackets tan to the power of negative 1 end exponent straight y close square brackets subscript 0 superscript 1 space equals space 1 half open square brackets tan to the power of negative 1 end exponent 1 minus tan to the power of negative 1 end exponent 0 close square brackets space equals space 1 half open parentheses straight pi over 4 minus 0 close parentheses space equals space straight pi over 8$

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

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