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Integrals

Short Answer Type

121.
Evaluate: $integral subscript 0 superscript straight pi over 4 end superscript space fraction numerator sinx space cosx over denominator cos squared straight x plus sin to the power of 4 straight x end fraction dx.$

Let I = $integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sinx space cosx over denominator cos squared straight x plus sin to the power of 4 straight x end fraction dx space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 4 space sinx space cosx over denominator 2 space left parenthesis 2 space cos squared straight x right parenthesis space plus space left parenthesis 2 space sin squared straight x right parenthesis squared end fraction dx$

$equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 2 space sin 2 straight x over denominator 2 left parenthesis 1 plus cos 2 straight x right parenthesis plus left parenthesis 1 minus cos 2 straight x right parenthesis squared end fraction dx space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator 2 space sin 2 straight x over denominator 3 plus cos squared straight x end fraction dx$

Put cos 2x =y, ∴ –2 sin 2x dx = dy ⇒ 2 sin 2x dx = dy
When x = 0, y = cos 0 = 1

When $straight x space equals space straight pi over 4 comma space space space straight y space equals space cos straight pi over 2 space equals space 0$
$therefore$ I = - $integral subscript 1 superscript 0 fraction numerator 1 over denominator 3 plus straight y squared end fraction dy space equals space minus integral subscript 1 superscript 0 fraction numerator 1 over denominator straight y squared plus left parenthesis square root of 3 right parenthesis squared end fraction dy space equals space minus fraction numerator 1 over denominator square root of 3 end fraction open square brackets tan to the power of negative 1 end exponent fraction numerator straight y over denominator square root of 3 end fraction close square brackets subscript 1 superscript 0$
$equals negative fraction numerator 1 over denominator square root of 3 end fraction open square brackets tan to the power of negative 1 end exponent 0 space minus tan to the power of negative 1 end exponent fraction numerator 1 over denominator square root of 3 end fraction close square brackets space equals space minus fraction numerator 1 over denominator square root of 3 end fraction open square brackets 0 minus straight pi over 6 close square brackets space equals space fraction numerator straight pi over denominator 6 square root of 3 end fraction$

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

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