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Integrals

Short Answer Type

191.

Show that:
$integral subscript straight pi over 4 end subscript superscript straight pi over 4 end superscript open vertical bar sin space straight x close vertical bar dx space equals space 2 minus square root of 2$

120 Views

Long Answer Type

192.

Show that:
$integral subscript negative 1 end subscript superscript 3 over 2 end superscript open vertical bar straight x space sinπ space straight x close vertical bar space dx space equals space 3 over straight pi plus 1 over straight pi squared$

152 Views

Short Answer Type

193.

Show that:
$integral subscript 0 superscript 2 straight x square root of 2 minus straight x end root space equals fraction numerator 16 square root of 2 over denominator 15 end fraction$

147 Views

194.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript 1 space straight x space left parenthesis 1 minus straight x right parenthesis to the power of straight n space dx$

108 Views

195.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx minus cosx over denominator 1 plus sinxcosx end fraction dx$

134 Views

196.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript 2 straight x space log space tanx space dx space equals space 0$

110 Views

197.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 2 space log space sinx space minus space log space sin space 2 straight x right parenthesis space dx$

410 Views

198.

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx space plus space cosx end fraction dx space equals space straight pi squared over 4$

100 Views

199.
Show that:
$integral subscript 0 superscript infinity log space open parentheses straight x plus 1 over straight x close parentheses. space fraction numerator dx over denominator 1 plus straight x squared end fraction space equals space straight pi space log space 2$

Let I = $integral subscript 0 superscript infinity log space open parentheses straight x plus 1 over straight x close parentheses fraction numerator dx over denominator 1 plus straight x squared end fraction$
Put x = tan θ so that dx = sec² θ dθ
When $straight x equals 0 comma space space straight theta space equals space 0 comma space space space When space straight x rightwards arrow space infinity comma space space space straight theta rightwards arrow space straight pi over 2$
$therefore space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript log open parentheses tanθ plus 1 over tanθ close parentheses fraction numerator sec squared straight theta over denominator 1 plus tan squared straight theta end fraction dθ$

$equals space integral subscript 0 superscript straight pi over 2 end superscript log space open parentheses fraction numerator sin squared straight theta plus cos squared straight theta over denominator sinθ space cosθ end fraction close parentheses dθ space equals space integral subscript 0 superscript straight pi over 2 end superscript log space open parentheses fraction numerator 1 over denominator sin space straight theta space cosθ end fraction close parentheses dθ equals negative integral subscript 0 superscript straight pi over 2 end superscript log space sinθ space dθ space minus space integral subscript 0 superscript straight pi over 2 end superscript log space cos space straight theta space dθ space equals space minus open parentheses negative straight pi over 2 log space 2 close parentheses space minus open parentheses negative straight pi over 2 log space 2 close parentheses equals straight pi over 2 log space 2 space plus straight pi over 2 log space 2 space equals space straight pi space log space 2$

132 Views

200.

Show that:
$integral subscript 0 superscript 1 fraction numerator log space left parenthesis 1 plus straight x right parenthesis over denominator 1 plus straight x squared end fraction space equals space straight pi over 8 space log space 2$