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Integrals

Short Answer Type

171.

Evaluate $integral subscript negative 8 end subscript superscript 8 space log space open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses dx.$

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172.

Evaluate $integral subscript negative 2 end subscript superscript 2 space log space open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses space dx.$

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173.

Show that
$integral subscript 0 superscript straight pi over 2 end superscript space sin squared straight x space dx space equals space straight pi over 4$

102 Views

174.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript 2 straight pi end superscript cos to the power of 5 straight x space dx$

101 Views

175.

Show that:
$integral subscript negative straight pi over 2 end subscript superscript straight pi over 4 end superscript space sin to the power of 7 straight x space dx space equals space 0$

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176.
Show that:
$integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript straight x cubed space space cos cubed straight x space dx space equals 0$

Let I = $integral subscript negative straight pi divided by 4 end subscript superscript straight pi divided by 4 end superscript straight x cubed space cos cubed straight x space dx$
Let f(x) = $straight x cubed space cos cubed straight x$
$therefore space space space space space straight f left parenthesis negative straight x right parenthesis space equals space left parenthesis negative straight x right parenthesis cubed space cos cubed left parenthesis negative straight x right parenthesis space equals space straight x cubed cos cubed straight x space equals space minus straight f left parenthesis straight x right parenthesis therefore space space space straight f left parenthesis straight x right parenthesis space is space an space odd space function space of space straight x therefore space space space space space straight I space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space integral subscript negative straight a end subscript superscript straight a straight f left parenthesis straight x right parenthesis space dx space space equals space 0 space if space straight f left parenthesis straight x right parenthesis space is space an space odd space function close square brackets$

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Long Answer Type

177.

Evaluate $integral subscript 2 superscript 3 fraction numerator square root of straight x over denominator square root of 5 minus straight x end root plus square root of straight x end fraction dx$

92 Views

Short Answer Type

178.

Evaluate $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx comma space space space where space space space straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell 4 straight x plus 3 space if space 1 space less or equal than space straight x space less or equal than space 2 end cell row cell 3 straight x plus 5 space if space 2 space less or equal than space straight x space less or equal than space 4 end cell end table close.$

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179.

Evaluate $integral subscript 0 superscript straight pi over 4 end superscript space log space left parenthesis 1 plus tanx right parenthesis space dx.$