Evaluate the definite integral _2^3 xdx x^2 + 1 - Vidyadip

Question

Evaluate the definite integral $\int\limits_2^3 {\frac{{xdx}}{{{x^2} + 1}}} $

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Answer

$\int\limits_2^3 {\frac{{xdx}}{{{x^2} + 1}}} = \frac{1}{2}\int\limits_2^3 {\frac{{2x}}{{{x^2} + 1}}dx} $

$= \frac{1}{2}\left( {\log \left| {{x^2} + 1} \right|} \right)_2^3$

$= \frac{1}{2}\left( {\log \left| {10} \right| - \log \left| 5 \right|} \right)$

$= \frac{1}{2}\left( {\log 10 - \log 5} \right)$

$ = \frac{1}{2}\log \frac{{10}}{5}$

$= \frac{1}{2}\log 2$

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