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101.
Evaluate the following integrals
$integral subscript negative 1 end subscript superscript 1 space straight x cubed space left parenthesis straight x to the power of 4 plus 1 right parenthesis cubed space dx$

Let I = $integral subscript negative 1 end subscript superscript 1 straight x cubed left parenthesis straight x to the power of 4 plus 1 right parenthesis cubed dx space equals space integral subscript negative 1 end subscript superscript 1 left parenthesis straight x to the power of 4 plus 1 right parenthesis cubed space straight x cubed dx$
Put $straight x to the power of 4 plus 1 space equals space straight y comma space space space space space therefore space space space space 4 space straight x cubed space dx space space equals dy space space space space space space rightwards double arrow space space space space straight x cubed dx space equals space 1 fourth dy$
When x = – 1, y = 1 + 1 = 2 When x = 1, y = 1 + 1 = 2
$therefore space space space space space straight I space equals space 1 fourth integral subscript 2 superscript 2 straight y cubed dy space equals 1 fourth open square brackets straight y to the power of 4 over 4 close square brackets subscript 2 superscript 2 space equals space 1 over 16 open square brackets straight y to the power of 4 close square brackets subscript 2 superscript 2 space equals space 1 over 16 left square bracket 16 minus 16 right square bracket space equals space 1 over 16 cross times 0 space equals space 0$

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