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Integrals

Short Answer Type

251.

Find space integral fraction numerator dx over denominator straight x squared space plus 4 straight x space plus 8 end fraction

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252.

Find $integral fraction numerator 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis squared end fraction dx$

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253.
Evaluate: $integral subscript 0 superscript straight pi fraction numerator straight x space sin space straight x over denominator 1 plus cos squared x end fraction dx$

$integral subscript 0 superscript straight pi fraction numerator straight x space sin space straight x over denominator 1 plus cos squared x end fraction dx space space space...... space left parenthesis straight i right parenthesis open square brackets By space using space integral subscript 0 superscript straight a space straight f left parenthesis straight x right parenthesis dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis dx close square brackets we space get space equals space integral subscript 0 superscript straight pi fraction numerator left parenthesis straight pi minus straight x right parenthesis space sin space left parenthesis straight pi minus straight x right parenthesis over denominator 1 plus cos squared space left parenthesis straight pi minus straight x right parenthesis end fraction dx space space equals space integral subscript 0 superscript straight pi fraction numerator left parenthesis straight pi minus straight x right parenthesis space sin space straight x over denominator 1 plus cos squared space straight x end fraction dx space space....... space left parenthesis ii right parenthesis$
Adding (i) and (ii) we get
$2 straight I space equals space integral subscript 0 superscript straight pi space fraction numerator straight x space sin space straight x over denominator 1 space plus space cos squared straight x end fraction space dx space plus space integral subscript 0 superscript straight pi fraction numerator left parenthesis straight pi minus straight x right parenthesis space sin space straight x over denominator 1 plus space cos squared space straight x end fraction dx space equals space integral subscript 0 superscript straight pi space fraction numerator left parenthesis straight x space plus straight pi space minus straight x right parenthesis space Sin space straight x over denominator 1 plus space cos squared straight x end fraction space dx 2 straight I space equals space integral subscript 0 superscript straight pi fraction numerator straight pi space sin space straight x over denominator 1 space space plus space cos squared straight x end fraction space dx straight I space equals space integral subscript 0 superscript straight pi straight pi over 2 fraction numerator space sin space straight x over denominator 1 plus cos squared space straight x space end fraction dx$

Let cos x = t,

-sin xdx = dt
also, x = 0, t = cos 0 = 1 and x = π, t= cos π=-1
$straight I space equals space straight pi over 2 space integral subscript 1 superscript negative 1 end superscript space fraction numerator negative 1 over denominator 1 plus straight t squared end fraction dt straight I space equals space minus straight pi over 2 open square brackets tan to the power of negative 1 end exponent space straight t close square brackets subscript 1 superscript negative 1 end superscript space equals space minus straight pi over 2 open square brackets tan to the power of negative 1 end exponent space left parenthesis negative 1 right parenthesis minus tan to the power of negative 1 end exponent space left parenthesis 1 right parenthesis close square brackets space equals space minus straight pi over 2 open square brackets negative straight pi over 4 minus straight pi over 4 close square brackets space equals straight pi over 2 space straight x space fraction numerator 2 straight pi over denominator 4 end fraction straight I space equals space straight pi squared over 4$