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

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

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Short Answer Type

178.
Evaluate $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx comma space space space where space space space straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell 4 straight x plus 3 space if space 1 space less or equal than space straight x space less or equal than space 2 end cell row cell 3 straight x plus 5 space if space 2 space less or equal than space straight x space less or equal than space 4 end cell end table close.$

Here $straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell 4 straight x plus 3 space if space 1 space less or equal than space straight x space less or equal than 2 end cell row cell 3 straight x plus 5 space if space 2 space less or equal than space straight x space less or equal than 4 end cell end table close$
Let I = $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 1 superscript 2 straight f left parenthesis straight x right parenthesis space dx space plus space integral subscript 2 superscript 4 straight f left parenthesis straight x right parenthesis space dx$
$equals space integral subscript 1 superscript 2 left parenthesis 4 straight x plus 3 right parenthesis space dx plus integral subscript 2 superscript 4 space left parenthesis 3 straight x plus 5 right parenthesis space dx space equals space open square brackets 4 straight x squared over 2 plus 3 space straight x close square brackets subscript 1 superscript 2 plus open square brackets fraction numerator 3 straight x squared over denominator 2 end fraction plus 5 straight x close square brackets subscript 2 superscript 4 equals space open square brackets left parenthesis 8 plus 6 right parenthesis space minus space left parenthesis 2 plus 3 right parenthesis close square brackets space plus space open square brackets left parenthesis 24 plus 20 right parenthesis space minus space left parenthesis 6 plus 10 right parenthesis close square brackets equals space left parenthesis 14 minus 5 right parenthesis space plus space left parenthesis 44 minus 16 right parenthesis space equals space 9 space plus space 28 space equals space 37.$

97 Views

179.

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