Evaluate the following integral:
$integral subscript negative 1 end subscript superscript 1 fraction numerator dx over denominator straight x squared plus 2 straight x plus 5 end fraction$

Solution

Let I = $integral subscript negative 1 end subscript superscript 1 fraction numerator 1 over denominator straight x squared plus 2 straight x plus 5 end fraction dx space equals space integral subscript negative 1 end subscript superscript 1 fraction numerator 1 over denominator left parenthesis straight x squared plus 2 straight x plus 1 right parenthesis plus 4 end fraction dx$
$equals space integral subscript negative 1 end subscript superscript 1 fraction numerator 1 over denominator left parenthesis straight x plus 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared end fraction dx space equals space 1 half open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 1 over denominator 2 end fraction close parentheses close square brackets subscript negative 1 end subscript superscript 1 equals space 1 half left square bracket tan to the power of negative 1 end exponent 1 space minus space tan to the power of negative 1 end exponent 0 right square bracket space equals space 1 half open parentheses straight pi over 4 minus 0 close parentheses space equals space straight pi over 8$

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Integrals

Evaluate the following integral:
$integral subscript negative 1 end subscript superscript 1 fraction numerator dx over denominator straight x squared plus 2 straight x plus 5 end fraction$

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