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Integrals

Short Answer Type

101.

Evaluate the following integrals
$integral subscript negative 1 end subscript superscript 1 space straight x cubed space left parenthesis straight x to the power of 4 plus 1 right parenthesis cubed space dx$

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102.

Evaluate the following integrals
$integral subscript 0 superscript 1 straight x square root of 1 minus straight x squared end root dx$

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103.

Evaluate the following integrals:
$integral subscript negative 1 end subscript superscript 1 fraction numerator 5 straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx$

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104.

Evaluate the following integral:
$integral subscript 0 superscript 1 fraction numerator 5 straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx$

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Long Answer Type

105.

Evaluate the following integral:
$integral subscript 0 superscript 2 fraction numerator dx over denominator straight x plus 4 minus straight x squared end fraction$

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Short Answer Type

106. Evaluate the following integral:

integral subscript negative 1 end subscript superscript 1 fraction numerator dx over denominator straight x squared plus 2 straight x plus 5 end fraction

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107.

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript space square root of sin space straight ϕ end root space cos to the power of 5 straight ϕ space dϕ space$

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Long Answer Type

108.

Evaluate $integral subscript 0 superscript 2 fraction numerator 5 straight x plus 1 over denominator straight x squared plus 4 end fraction dx.$

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109.

Prove that: $integral subscript negative straight a end subscript superscript straight a square root of fraction numerator straight a minus straight x over denominator straight a plus straight x end fraction end root space dx space equals space straight a space straight pi.$

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110.
Evaluate $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sinx space cosx end root end fraction dx.$

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx plus cosx over denominator square root of sin space straight x space cosx end root end fraction dx space equals space 4 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx plus cosx over denominator square root of 2 space sinx space cosx end root end fraction dx$

$equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sin space 2 straight x end root end fraction dx space equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx space plus space cosx over denominator square root of 1 minus left parenthesis 1 minus sin space 2 straight x right parenthesis end root end fraction dx$
$equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of 1 minus left parenthesis sin squared straight x plus cos squared straight x minus 2 sinx space cosx right parenthesis end root end fraction dx equals space 4 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x space over denominator square root of 1 minus left parenthesis sinx space space minus cosx right parenthesis squared end root end fraction dx$
Put sin x - cos x = y, ∴ (cos x + sin x) dx = dy When x = 0, y = sin 0 – cos 0 = 0 – 1 = – 1
When $straight x equals space straight pi over 2 comma space space space straight y space equals space sin straight pi over 2 minus cos straight pi over 2 space equals space 1 minus 0 space equals space 1$
$therefore$ $straight I space equals space 4 integral subscript negative 1 end subscript superscript 1 fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction dy space equals space 4 open square brackets sin to the power of negative 1 end exponent space straight y close square brackets subscript negative 1 end subscript superscript 1 space equals space 4 open square brackets sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis space minus space sin to the power of negative 1 end exponent left parenthesis negative 1 right parenthesis close square brackets$
$equals space 4 space open square brackets straight pi over 2 minus open parentheses negative straight pi over 2 close parentheses close square brackets space equals space 4 open parentheses straight pi over 2 plus straight pi over 2 close parentheses space equals space 4 straight pi$