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Integrals

Short Answer Type

201.

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

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

Let I = $integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator straight theta over denominator sin space straight theta end fraction close parentheses squared space dθ space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator straight theta squared over denominator sin squared space straight theta end fraction dθ space equals space integral subscript 0 superscript straight pi over 2 end superscript straight theta squared space cosec squared straight theta space dθ$
$equals space open square brackets straight theta squared left parenthesis negative cot space straight theta right parenthesis close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space integral subscript 0 superscript straight pi over 2 end superscript 2 space straight theta. space left parenthesis negative cot space straight theta right parenthesis space dθ space equals space open square brackets straight theta squared space left parenthesis negative cot space straight theta right parenthesis close square brackets subscript 0 superscript straight pi over 2 end superscript plus 2 integral subscript 0 superscript straight pi over 2 end superscript straight theta. space cot space straight theta space dθ space equals space open parentheses negative straight pi squared over 4 cot straight pi over 2 plus 0 close parentheses plus 2 open curly brackets open square brackets straight theta space. log space sinθ close square brackets subscript 0 superscript straight pi over 2 end superscript minus integral subscript 0 superscript straight pi over 2 end superscript 1. space log space sinθ space dθ close curly brackets equals space straight pi squared over 4 cross times 0 plus 0 plus 2 open square brackets straight theta. space log space sinθ close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space 2 integral subscript 0 superscript straight pi over 2 end superscript log space sinθ space dθ space equals 2 open square brackets straight pi over 2 log space sin straight pi over 2 minus 0 close square brackets space minus space 2 open parentheses negative straight pi over 2 log space 2 close parentheses space equals space 2 space left square bracket 0 minus 0 right square bracket space plus space straight pi space log space 2 space space equals space straight pi space log space 2$

153 Views

210.

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