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

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254.
Evaluate: $integral subscript 0 superscript 3 divided by 2 end superscript vertical line space straight x space sin space πx vertical line space dx$

$integral subscript 0 superscript 3 divided by 2 end superscript vertical line space straight x space sin space πx vertical line space dx$
For 0 < x < 1, x sinπx dx
For 1 < x <3/2, x sinπx< 0
Applying these in equation
$integral subscript 0 superscript 3 divided by 2 end superscript vertical line straight x space sin space πx vertical line space dx space equals space integral subscript 0 superscript 1 space straight x space sin space πdx space minus space integral subscript 1 superscript 3 divided by 2 end superscript straight x space sin space straight pi space straight x space dx Let space straight I space equals space integral straight x space sin space straight pi space straight x space dx space equals space straight x integral sinπ space straight x space dx space minus space integral open parentheses straight d over dx straight x integral sin space straight pi space straight x space dx close parentheses dx space equals space straight x open parentheses negative fraction numerator cos space πx over denominator straight pi end fraction close parentheses minus integral open parentheses fraction numerator negative cos space straight pi space straight x over denominator straight pi end fraction close parentheses space dx space equals space fraction numerator negative straight x space cos space straight pi space straight x over denominator straight pi end fraction space plus space fraction numerator sin space straight pi space straight x over denominator straight pi squared end fraction$

Now applying the limits we get,
$integral subscript 0 superscript 3 divided by 2 end superscript vertical line space straight x space sin space straight pi space straight x vertical line space dx space equals space open square brackets fraction numerator negative straight x space cos space straight pi space straight x over denominator straight pi end fraction space plus fraction numerator sin space straight pi space straight x over denominator straight pi squared end fraction close square brackets subscript 0 superscript 1 space minus space open square brackets fraction numerator negative straight x space cos space straight pi space straight x over denominator straight pi end fraction space plus fraction numerator sin space straight pi space straight x over denominator straight pi squared end fraction close square brackets subscript 1 superscript 3 divided by 2 end superscript open square brackets open parentheses fraction numerator negative cos space straight pi over denominator straight pi end fraction plus fraction numerator sin space straight pi over denominator straight pi squared end fraction close parentheses minus left parenthesis 0 plus 0 right parenthesis close square brackets minus open square brackets open parentheses fraction numerator negative begin display style 3 over 2 end style space cos begin display style fraction numerator 3 straight pi over denominator 2 end fraction end style over denominator straight pi end fraction space plus fraction numerator sin begin display style fraction numerator 3 straight pi over denominator 2 end fraction end style over denominator straight pi squared end fraction close parentheses space minus space open parentheses fraction numerator negative cos space straight pi over denominator straight pi end fraction plus fraction numerator sin space straight pi over denominator straight pi squared end fraction close parentheses close square brackets open square brackets open parentheses 1 over straight pi plus 0 close parentheses close square brackets minus open square brackets open parentheses 0 minus 1 over straight pi squared close parentheses minus open parentheses 1 over straight pi plus 0 close parentheses close square brackets space equals space 1 over straight pi space plus 1 over straight pi squared space plus 1 over straight pi space equals space 2 over straight pi space plus space 1 over straight pi squared space equals space fraction numerator 2 straight pi space plus 1 over denominator straight pi squared end fraction$