Solve Int. xe^xlnx Difficult Integral

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SUMMARY

The integral $$\int xe^x \ln x \, dx$$ can be evaluated using integration by parts. The result is expressed as $$(x-1)e^x \ln x - e^x + \text{Ei}(x) + c$$, where $\text{Ei}(x)$ represents the Exponential Integral Function. This method effectively simplifies the integral by breaking it down into manageable components, demonstrating the utility of integration by parts in solving complex integrals.

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Can anyone help me Evaluating this integral!
$$\int xe^xlnxdx$$
 
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Maged Saeed said:
Can anyone help me Evaluating this integral!
$$\int xe^xlnxdx$$

Integrating by parts You obtain...

$\displaystyle \int x\ e^x\ \ln x\ dx = (x-1)\ e^{x}\ \ln x - \int \frac{x-1}{x}\ e^{x}\ dx = (x-1)\ e^{x}\ \ln x - e^{x} + \text{Ei}\ (x) + c\ (1)$

... where $\text{Ei} (x)$ is the 'Exponential Integral Function'...

Kind regards

$\chi$ $\sigma$
 
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