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  4. The value of ∫1e (1+x2 ln x/x+x2 ln x) d x is

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Q. The value of $\int_{1}^{e} \frac{1+x^{2} \ln x}{x+x^{2} \ln x} d x$ is

Integrals

Solution:

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