1. ## Integration

This my big problem in my integration problem, $\displaystyle \int{e^x \ln x$.In this case i choose u=$\displaystyle \ln x$ and $\displaystyle \frac{de^x} {dx}=e^x$.Then i stuck on this $\displaystyle \int{\frac{e^x}{x}}$.please give me a solution

2. ## Re: Integration

I agree with your method of integration by parts, so you should end up with \displaystyle \displaystyle \begin{align*} \int{e^x\ln{(x)}\,dx} = e^x\ln{(x)} - \int{\frac{e^x}{x}\,dx} \end{align*}. The second function does not have a solution in terms of the elementary functions, but Wolfram tells us that this is the Exponential Integral.

3. ## Re: Integration

Of course, a series solution could be appropriate...

\displaystyle \displaystyle \begin{align*} e^x &= \sum_{n = 0}^{\infty}{\frac{x^n}{n!}} \\ \frac{e^x}{x} &= \sum_{n = 0}^{\infty}{\frac{x^{n-1}}{n!}} \\ &= \frac{1}{x} + \sum_{n = 1}^{\infty}{\frac{x^{n-1}}{n!}} \\ \int{\frac{e^x}{x}\,dx} &= \int{\frac{1}{x} + \sum_{n=1}^{\infty}{\frac{x^{n-1}}{n!}}\,dx} \\ &= \ln{|x|} + \sum_{n = 1}^{\infty}{\frac{x^n}{n \cdot n!}} + C \end{align*}

4. ## Re: Integration

If there is no other way to approach this problem,like let u=e^x and d[xlnx-x]/dx=lnx.

5. ## Re: Integration

Originally Posted by srirahulan
If there is no other way to approach this problem,like let u=e^x and d[xlnx-x]/dx=lnx.
No, that gives you a more difficult integral to try to solve. Not all integrals have closed-form solutions in terms of elementary functions.