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

Let I = $integral subscript 0 superscript 1 fraction numerator 5 straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx space equals space 5 integral subscript 0 superscript 1 fraction numerator straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx$
Put $4 plus straight x squared space equals space straight y semicolon space space space space space therefore space 2 space straight x space dx space equals space dy space rightwards double arrow space space space straight x space dx space equals space 1 half dy$
When x = 0, y = 4 + 0 = 4
$therefore$ I = $5 over 2 integral subscript 4 superscript 5 1 over straight y squared dy space equals space 5 over 2 integral subscript 2 superscript 5 straight y to the power of negative 2 end exponent dy space equals space 5 over 2 open parentheses fraction numerator straight y to the power of negative 1 end exponent over denominator negative 1 end fraction close parentheses subscript 4 superscript 5$
$equals negative 5 over 2 open square brackets 1 over straight y close square brackets subscript 4 superscript 5 space equals space minus 5 over 2 open square brackets 1 fifth minus 1 fourth close square brackets space equals space minus 5 over 2 open square brackets fraction numerator 4 minus 5 over denominator 20 end fraction close square brackets equals negative 5 over 2 cross times fraction numerator negative 1 over denominator 20 end fraction equals space 1 over 8$

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