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

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

97 Views

179.
Evaluate $integral subscript 0 superscript straight pi over 4 end superscript space log space left parenthesis 1 plus tanx right parenthesis space dx.$

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space log left parenthesis 1 plus tanx right parenthesis space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript log open square brackets 1 plus tan open parentheses straight pi over 4 minus straight x close parentheses close square brackets dx$
   $open square brackets because space space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space log open square brackets 1 plus fraction numerator 1 minus tanx over denominator 1 plus tanx end fraction close square brackets dx space equals space integral subscript 0 superscript straight pi over 4 end superscript space log space open square brackets fraction numerator 1 plus tanx plus 1 minus tanx over denominator 1 plus tanx end fraction close square brackets dx$

   $equals space integral subscript 0 superscript straight pi over 4 end superscript log open parentheses fraction numerator 2 over denominator 1 plus tanx end fraction close parentheses dx space equals space integral subscript 0 superscript straight pi over 4 end superscript log space 2 space dx space minus space integral subscript 0 superscript straight pi over 4 end superscript log left parenthesis 1 plus tanx right parenthesis space dx$

$therefore space space space straight I space equals space log space 2 integral subscript 0 superscript straight pi over 4 end superscript space 1 space dx space minus space 1$ $open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
$rightwards double arrow space space space 2 space straight I space equals space log space 2 space open square brackets straight x close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space log space 2 open square brackets straight pi over 4 minus 0 close square brackets space space space space space space rightwards double arrow space space space 2 space straight I space equals space straight pi over 4 log 2 space rightwards double arrow space straight I space equals space straight pi over 8 log space 2$