# Integral e^x.(lnx)dx=?

1. Aug 17, 2008

### sumerman

i have tried some methods like uv - integral vdu but can't reach the answer

2. Aug 17, 2008

### arildno

You won't be able to do it.

3. Aug 17, 2008

4. Jul 3, 2010

### tony.c.tan

Try to take the derivative of this with respect to $x$, and see what do you get:

$$e^x\left[\ln x-\sum_{i=1}^{\infty}(i-1)!x^{-i}\right]$$

Last edited: Jul 3, 2010
5. Jul 3, 2010

cute.

6. Jul 3, 2010

### arildno

Supercool!

Would you show what techniques are useful to get that anti-derivative?

7. Jul 3, 2010

### g_edgar

Woh! a series that diverges for every x ... what a useful answer ...

8. Jul 4, 2010

### arildno

Yeah, i noticed that after a while...

It MIGHT be, that the formula CAN be used, with extreme caution, since we will mainly use differences between two "values" of the anti-derivative. Thos difference might be convergent, even though both terms are not.

But, then again, a numerical integration scheme might do equally well...

Last edited: Jul 4, 2010
9. Jun 25, 2012

### norice4u

apologize for my ignorance but what is that process called and the Sigma looking symbol? I am an Yr 12 student currently doing VCE and studying specialist math and is just stumped on an equation hoping to find an answer in here.

NB: excuse me for fail to type with mathematic symbol
Anti-Differentiate x^x=?

But really i am asking how to anti-differentiate x^x(lnx+1) which comes from the derivative of y=x^x
Because out of curiosity i always hold the belief in math if there is a forward operation there should be a backwards operation so if i can differentiate x^x to get that ugly function to anti-differentitate what operations would i have to undergo.

Thanks for the trouble of reading this passage.

10. Jun 25, 2012

### micromass

Staff Emeritus

The Sigma symbol is the summation symbol. It's just the shorthand for a sum. For example

$$\sum_{i=1}^3 i^2= 1^2+2^2+3^2$$

Of course, things like $\sum_{i=1}^{+\infty}$ can not be defined as such since the sum would be infinite. Infinite sums are called series in mathematics and have a very big underlying theory.

This function certainly has an anti-derivative, but it can not be written in terms of elementary functions. Most functions do not have elementary anti-derivatives.

This integral can be solved by an easy substitution.

The "backwards operation" of differentiation is called integration. But integration is much more harder. Where differentiation has nice algorithms which can be used to differentiate all nice functions, the same is not true for integration. Most integrals are not easy to solve.