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

Let I = $integral subscript 0 superscript straight pi over 4 end superscript fraction numerator sinx plus cosx over denominator 9 plus 16 sin 2 straight x end fraction dx$
Put sin x – cos x = t, ∴ (cos x + sin x) dx = dt
Also sin² x + cos² x – 2 sin x cos x = t² ⇒ 1 – sin 2 x = t² When x = 0, t = sin 0 – cos 0 = 0 – 1= – 1

When

straight x equals space straight pi over 4 comma space space space straight t space equals space sin straight pi over 4 minus cos straight pi over 4 space equals space fraction numerator 1 over denominator square root of 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction equals 0

straight I space equals space integral subscript negative 1 end subscript superscript 0 fraction numerator 1 over denominator 9 plus 16 left parenthesis 1 minus straight t squared right parenthesis end fraction dt space equals space integral subscript negative 1 end subscript superscript 0 fraction numerator 1 over denominator 25 minus 16 straight t squared end fraction dt

equals space 1 over 16 integral subscript negative 1 end subscript superscript 0 space fraction numerator 1 over denominator open parentheses begin display style 5 over 4 end style close parentheses squared minus straight t squared end fraction space dt space equals space 1 over 16 cross times fraction numerator 1 over denominator 2 cross times begin display style 5 over 4 end style end fraction open square brackets log space open vertical bar fraction numerator begin display style 5 over 4 end style plus straight t over denominator begin display style 5 over 4 end style minus straight t end fraction close vertical bar close square brackets subscript negative 1 end subscript superscript 0 equals space 1 over 40 open parentheses log space 1 space minus space log space 1 over 9 close parentheses space equals space 1 over 40 left parenthesis log space 1 space minus space log space 1 space plus space log space 9 right parenthesis space equals space 1 over 40 space log space 9.

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