_ 1 2 ^ 2 x^2ln x (1+x^2)^3 dx - Vidyadip

Question

$\int_{\frac{1}{2}}^{2} \frac{x^2ln x}{(1+x^2)^3}dx$

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Answer

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$\int\limits_{\frac{1}{2}}^2 {\frac{{{x^2}\ln x}}{{{{\left( {1 + {x^2}} \right)}^3}}}} dx = \int\limits_{\frac{1}{2}}^1 {\frac{{{x^2}\ln x}}{{{{\left( {1 + {x^2}} \right)}^3}}}} dx + \int\limits_1^2 {\frac{{{x^2}\ln x}}{{{{\left( {1 + {x^2}} \right)}^3}}}dx} = {I_1} + {I_2}$

for $\mathrm{I}_{1},$ put $\mathrm{x}=\frac{1}{\mathrm{t}}$ we get $\mathrm{I}_{1}=-\mathrm{I}_{2}$

Given integral $=0$

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