Question

$\int [{\left(\frac{x}{e}\right)}^x+{\left(\frac{e}{x}\right)}^x]lnx$ dx

Answer 1

$\int [{\left(\frac{x}{e}\right)}^x+{\left(\frac{e}{x}\right)}^x]lnx$ dx

Let ${\left(\frac{x}{e}\right)}^x=t$

$x\ln \left(\frac{x}{e}\right)=\ln t$

$x\ln x-x=\ln t$

${\left(\frac{e}{x}\right)}^x=\frac{1}{t}$

$x\ln \left(\frac{e}{x}\right)=-\ln t$

$x-x\ln x=-\ln t$

also

$(\ln x+\frac{x}{x}-1)dx=\frac{1}{t}dt$

$\ln xdx=\frac{1}{t}dt$

$\Rightarrow \int [t+\frac{1}{t}]\frac{1}{t}dt$

$\Rightarrow \int [\frac{1}{t}+\frac{1}{t^2}]dt$

$\Rightarrow \ln t-\frac{1}{t}+C$

$\Rightarrow x\ln \left(\frac{e}{x}\right)-{\left(\frac{e}{x}\right)}^x+C$