Question

$\int \frac{e^x}{x}\left(1+x.lnx\right)dx$

Answer 1

$\int \frac{e^x}{x}(1+x\ln x)dx$

$=\int \frac{e^x}{x}dx+\int \frac{e^x}{x}x\ln xdx$

$=\int \frac{e^x}{x}dx+\int e^x\ln xdx$

Consider $\int \frac{e^x}{x}dx$

Integrating by parts,

Let $u=e^x\Rightarrow du=e^{x}dx$

$dv=\frac{1}{x}dx\Rightarrow v=\ln x$

$=e^x\ln x-\int \ln x{e}^{x}dx$

$\therefore \int \frac{e^x}{x}dx=e^x\ln x-\int \ln x{e}^{x}dx+c$

$\therefore \int \frac{e^x}{x}dx+\int e^x\ln xdx$

$=e^x\ln x-\int \ln x{e}^{x}dx+\int \ln x{e}^{x}dx+c$

$=e^x\ln x+c$ where $c$ is the constant of integration.