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Integrals

Short Answer Type

171.
Evaluate $integral subscript negative 8 end subscript superscript 8 space log space open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses dx.$

integral subscript negative straight a end subscript superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space minus integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space plus space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx

rightwards double arrow space space space integral subscript negative straight a end subscript superscript straight a space straight f left parenthesis straight x right parenthesis space dx space equals space 0

Let

straight I space equals space 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

Let

straight f left parenthesis straight x right parenthesis space equals space log open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses

therefore space space space straight f left parenthesis negative straight x right parenthesis space equals space log open parentheses fraction numerator 2 minus straight x over denominator 2 plus straight x end fraction close parentheses space equals space log open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses to the power of negative 1 end exponent space equals space minus log open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses space equals space minus straight f left parenthesis straight x right parenthesis

therefore space space space space space space straight f left parenthesis straight x right parenthesis

is an odd function of x.

therefore space space space space space straight I equals space 0

open square brackets because space integral subscript straight a superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space 0 space if space straight f left parenthesis straight x right parenthesis space is space an space odd space function space of space straight x close square brackets

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

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Long Answer Type

177.

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

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