MHB Special Funcs: Evaluating $$\int \frac{1}{\sqrt{x}\ln(x)}$$

[Amad27]

Hi,

Recently, I had stumbled across:

$$\int \frac{1}{\sqrt{x}\ln(x)}$$
Let $f(x) = \frac{1}{\sqrt{x}\ln(x)}$

I noticed there is no elementary antiderivative. I want to evaluate this using special functions, but as of right now, I would like some advice as I have no clue about special functions.

Let $u = \sqrt{x}$

$$ = \int \frac{2}{ln(u^2)} du$$

This is where it gets interesting, there is no elementary antiderivative, then how can we evaluate this in terms of special functions.

And AFTER THAT I want to experiment with

$$\int_{2}^{3} f(x) \,dx$$

thanks!

 

[chisigma]

Hi,

Recently, I had stumbled across:

$$\int \frac{1}{\sqrt{x}\ln(x)}$$
Let $f(x) = \frac{1}{\sqrt{x}\ln(x)}$

I noticed there is no elementary antiderivative. I want to evaluate this using special functions, but as of right now, I would like some advice as I have no clue about special functions.

Let $u = \sqrt{x}$

$$ = \int \frac{2}{ln(u^2)} du$$

This is where it gets interesting, there is no elementary antiderivative, then how can we evaluate this in terms of special functions.

And AFTER THAT I want to experiment with

$$\int_{2}^{3} f(x) \,dx$$

thanks!

Setting $\displaystyle x=e^{u}$ the integral becomes...

$\displaystyle \int \frac{d x}{\sqrt{x}\ \ln x} = \int \frac{e^{\frac{u}{2}}}{u}\ d u = \text{Ei}\ (\frac{u}{2}) + c\ (1)$

... so that your 'special function' is the Exponential Integral Function...

Kind regards

$\chi$ $\sigma$

 

[Euge]

Hi,

Recently, I had stumbled across:

$$\int \frac{1}{\sqrt{x}\ln(x)}$$
Let $f(x) = \frac{1}{\sqrt{x}\ln(x)}$

I noticed there is no elementary antiderivative. I want to evaluate this using special functions, but as of right now, I would like some advice as I have no clue about special functions.

Let $u = \sqrt{x}$

$$ = \int \frac{2}{ln(u^2)} du$$

This is where it gets interesting, there is no elementary antiderivative, then how can we evaluate this in terms of special functions.

And AFTER THAT I want to experiment with

$$\int_{2}^{3} f(x) \,dx$$

thanks!

Hello again Olok,

I suggest letting $u = \ln(x)$ instead, so that $x = e^u$ and $dx = e^u\, du$. Thus

$$ \int \frac{dx}{\sqrt{x}\ln(x)} = \int \frac{e^u\, du}{e^{u/2}\,u} = \int \frac{e^{u/2}}{u}\,du.$$

Letting $v = -u/2$, $dv/v = du/u$, and so

$$ \int \frac{e^{u/2}}{u}\, du = \int \frac{e^{-v}}{v}\, dv.$$

Although we're dealing with indefinite integrals, the last integral is closely related to the exponential integral,

$$\text{Ei}(x) = \int_x^\infty \frac{e^{-t}}{t}\, dt.$$

 

[alyafey22]

You can also take a look at Incomplete Gamma function.

 

[Amad27]

Hi,

So can we find the definite integral of that from

$2$ to $3$?