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Integrals

Short Answer Type

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

integral fraction numerator 1 over denominator straight x squared space plus 4 straight x space 8 end fraction dx straight I space equals space integral fraction numerator 1 over denominator straight x squared plus 4 straight x space plus 8 end fraction dx equals space integral fraction numerator 1 over denominator left parenthesis straight x plus 2 right parenthesis squared space plus 2 squared end fraction dx By space using space Integral space Formula integral fraction numerator 1 over denominator straight x squared space plus straight a squared end fraction space dx space equals space 1 over straight a tan to the power of negative 1 end exponent open parentheses straight x over straight a close parentheses equals 1 half tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator 2 end fraction close parentheses space plus straight C

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