How Do I Calculate the Integral of e^x(lnx) and Anti-Derivative of x^x?

[sumerman]

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

 

[arildno]

You won't be able to do it.

 

[Crosson]

The antiderivative of your function involves the exponential integral function, which is defined here:

http://mathworld.wolfram.com/ExponentialIntegral.html

 

[tony.c.tan]

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

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

 

[The Chaz]

cute.

 

[arildno]

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

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

Supercool!

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

 

[g_edgar]

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

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

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

 

[arildno]

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

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...

 

[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.

 

[micromass]

Please start a new thread next time instead of posting to an already existing thread.

apologize for my ignorance but what is that process called and the Sigma looking symbol?

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.

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

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.

But really i am asking how to anti-differentiate x^x(lnx+1) which comes from the derivative of y=x^x

This integral can be solved by an easy substitution.

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.

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.

Anyway, I am locking this thread. Please make a new thread if you have further questions.