Evaluate the following:
$integral subscript 1 superscript 2 open parentheses 1 over straight x minus fraction numerator 1 over denominator 2 straight x squared end fraction close parentheses space straight e to the power of 2 straight x end exponent dx$

Solution

Let
$straight I space equals space integral subscript 1 superscript 2 open parentheses 1 over straight x minus fraction numerator 1 over denominator 2 straight x squared end fraction close parentheses space straight e to the power of 2 straight x end exponent space dx$
$equals space integral subscript 1 superscript 2 1 over straight x straight e to the power of 2 straight x end exponent dx space plus space integral fraction numerator negative 1 over denominator 2 straight x squared end fraction straight e to the power of 2 straight x end exponent dx$
$equals space open square brackets 1 over straight x straight e to the power of 2 straight x end exponent over 2 close square brackets subscript 1 superscript 2 space minus space integral subscript 1 superscript 2 fraction numerator negative 1 over denominator straight x squared end fraction straight e to the power of 2 straight x end exponent over 2 dx plus integral subscript 1 superscript 2 fraction numerator negative 1 over denominator 2 straight x squared end fraction straight e to the power of 2 straight x end exponent dx$ [integrating by parts]
$equals space 1 half open square brackets straight e to the power of 2 straight x end exponent over straight x close square brackets subscript 1 superscript 2 space minus space integral subscript 1 superscript 2 fraction numerator negative 1 over denominator 2 straight x squared end fraction straight e to the power of 2 straight x end exponent dx plus integral subscript 1 superscript 2 fraction numerator negative 1 over denominator 2 space straight x squared end fraction straight e to the power of 2 straight x end exponent dx$
$equals space 1 half open square brackets straight e to the power of 4 over 2 minus straight e squared over 1 close square brackets space equals space 1 fourth left parenthesis straight e to the power of 4 minus 2 straight e squared right parenthesis space equals 1 fourth straight e squared left parenthesis straight e squared minus 2 right parenthesis$

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Integrals

Evaluate the following:
$integral subscript 1 superscript 2 open parentheses 1 over straight x minus fraction numerator 1 over denominator 2 straight x squared end fraction close parentheses space straight e to the power of 2 straight x end exponent dx$

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