|
|
$I=\int_{1}^{e}{\frac{2x^{2}+x(1+2\ln x) +\ln^{2}x}{(x^{2}+x\ln x)^{2}}}dx$
$I=\int_{1}^{e}{\frac{(x+\ln x)^2+x^2+x}{x^2(x+\ln x)^{2}}}dx$
$I=\int_{1}^{e}\left ( \frac{1}{x^2}+\frac{1+\frac{1}{x}}{(x+\ln x)^{2}}\right )dx$
$I=\int_{1}^{e} \frac{1}{x^2}dx +\int_{1}^{e} \frac{d(x+ \ln x)}{(x+\ln x)^{2}}$
$I=\left[ { -\frac{1}{x}-\frac{1}{x+\ln x}} \right]_{1}^{e}$
$I=\boxed{2-\frac{1}{e}-\frac{1}{e+1}}$
|