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

95 Views

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:
$integral subscript 0 superscript 2 straight pi end superscript cos to the power of 5 straight x space dx$

Let I = $integral subscript 0 superscript 2 straight pi end superscript cos to the power of 5 straight x space dx space equals space 2 integral subscript 0 superscript straight pi cos to the power of 5 straight x space dx$
$open square brackets because space space cos to the power of 5 space left parenthesis 2 straight pi minus straight x right parenthesis space equals space cos to the power of 5 straight x and space integral subscript 0 superscript 2 straight a end superscript straight f left parenthesis straight x right parenthesis space dx space equals space 2 integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space if space straight f left parenthesis 2 straight a minus straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis close square brackets equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets table row cell because space cos to the power of 5 left parenthesis straight pi minus straight x right parenthesis space equals space left parenthesis negative cos to the power of 5 straight x right parenthesis space equals space minus cos to the power of 5 straight x end cell row cell and space integral subscript 0 superscript 2 straight a end superscript straight f left parenthesis straight x right parenthesis space dx space equals space 0 space if space straight f left parenthesis 2 straight a minus straight x right parenthesis space equals space minus straight f left parenthesis straight x right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space$

101 Views

175.

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

102 Views

176.

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

94 Views

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

97 Views

179.

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