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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
$integral subscript 0 superscript 1 straight x square root of 1 minus straight x squared end root dx$

Let I = $integral subscript 0 superscript 1 straight x square root of 1 minus straight x squared end root space dx space equals space integral subscript 0 superscript 1 left parenthesis 1 minus straight x squared right parenthesis to the power of 1 half end exponent. space straight x space dx$
Put $1 minus straight x squared space equals space straight y comma space space space space space therefore space space space minus 2 space straight x space dx space equals space dy space space space space rightwards double arrow space space space straight x space dx space equals space minus 1 half dy$
When x = 0, y = 1 – 0 = 1 When x = 1, y = 1 – 1 = 0
$therefore$ I = $negative 1 half integral subscript 1 superscript 0 straight y to the power of 1 half end exponent dy space equals space minus 1 half open square brackets fraction numerator straight y to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 1 superscript 0 space equals space minus 1 third open square brackets straight y to the power of 3 over 2 end exponent close square brackets subscript 1 superscript 0 space equals space minus 1 third left square bracket 0 minus 1 right square bracket space equals space 1 third$

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