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Integrals

Short Answer Type

201.
Evaluate:
$integral subscript 0 superscript straight pi fraction numerator straight x space over denominator 1 plus sinx end fraction dx space equals space straight pi$

Let I = $integral subscript 0 superscript straight pi fraction numerator straight x over denominator 1 plus sinx end fraction dx$ ...(1)

$therefore space space space space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator straight pi minus straight x over denominator 1 plus sin left parenthesis straight pi minus straight x right parenthesis end fraction dx space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator straight pi minus straight x over denominator 1 plus sinx end fraction dx$

$rightwards double arrow space space space straight I space equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sinx end fraction dx minus integral subscript 0 superscript straight pi fraction numerator straight x over denominator 1 plus sinx end fraction dx$

$rightwards double arrow space space space space straight I space equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sinx end fraction dx minus 1$ $open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$

$rightwards double arrow space space space space 2 space straight I space equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sinx end fraction dx space equals straight pi integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sinx end fraction cross times fraction numerator 1 minus sinx over denominator 1 minus sinx end fraction dx$

$equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 minus sinx over denominator 1 minus sin squared straight x end fraction dx space equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 minus sinx over denominator cos squared straight x end fraction dx$

$equals space straight pi integral subscript 0 superscript straight pi open parentheses fraction numerator 1 over denominator cos squared straight x end fraction minus fraction numerator sinx over denominator cos squared straight x end fraction close parentheses dx space equals space straight pi integral subscript 0 superscript straight pi open parentheses sec squared straight x minus 1 over cosx. sinx over cosx close parentheses dx$

$equals space straight pi integral subscript 0 superscript straight pi left parenthesis sec squared straight x minus secx space tanx right parenthesis space dx space equals space straight pi space left square bracket tan space straight x space minus space sec space straight x right square bracket subscript 0 superscript straight pi$

$equals space straight pi left square bracket left parenthesis tan space straight pi space minus secπ right parenthesis space minus left parenthesis tan space 0 minus space sec space 0 right parenthesis right square bracket equals space straight pi left square bracket 0 minus left parenthesis negative 1 right parenthesis minus 0 plus 1 right square bracket space equals space straight pi left parenthesis 2 right parenthesis space equals space 2 straight pi$

$therefore space space 2 space straight I space equals space 2 space straight pi space space space rightwards double arrow space space space space straight I space equals space straight pi$

119 Views

202.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator straight x space dx over denominator sinx space plus space cosx end fraction space equals space fraction numerator straight pi over denominator 2 square root of 2 end fraction log left parenthesis square root of 2 plus 1 right parenthesis$

120 Views

Long Answer Type

203.

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

146 Views

Short Answer Type

204.

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

101 Views

205.

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx plus tanx end fraction dx space equals space straight pi open parentheses straight pi over 2 minus 1 close parentheses$

119 Views

Long Answer Type

206.

Show that:
$integral subscript 0 superscript 1 log space open parentheses 1 over straight x minus 1 close parentheses dx space equals space 0$

139 Views

Short Answer Type

207.

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

135 Views

208.

Show that:
$integral subscript 0 superscript straight pi space straight x space. space log space sinx space dx space equals space minus straight pi squared over 2 log space 2$

100 Views

209.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator straight theta over denominator sin space straight theta end fraction close parentheses squared space dθ space equals space straight pi space log space 2$

153 Views

210.

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space sinx over denominator 1 plus cos squared straight x end fraction dx space equals space straight pi squared over 4$