Question
समाकल $\int \limits_{1}^{e}\left\{\left(\frac{x}{e}\right)^{2 x}-\left(\frac{e}{x}\right)^{x}\right\} \log _{e} x d x$ होगा
$\operatorname{let}\left(\frac{x}{e}\right)^{2 x}=t,\left(\frac{e}{x}\right)^{x}=v$
$=\frac{1}{2} \int_{\left(\frac{1}{e}\right)^{2}}^{1} d t+\int_{e}^{1} d v=\frac{1}{2}\left(1-\frac{1}{e^{2}}\right)+(1-e)=\frac{3}{2}-\frac{1}{2 e^{2}}-e$
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