MHB Solve Int. xe^xlnx Difficult Integral

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The integral $\int xe^x \ln x \, dx$ can be evaluated using integration by parts. The result is $(x-1)e^x \ln x - \int \frac{x-1}{x} e^x \, dx$. This simplifies further to $(x-1)e^x \ln x - e^x + \text{Ei}(x) + c$, where $\text{Ei}(x)$ represents the Exponential Integral Function. The discussion highlights the steps involved in the integration process and the final expression for the integral. Understanding these steps is crucial for solving similar integrals effectively.
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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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