Integral w/ Unknown Table: Solve & Write in Closed-Form

[EngWiPy]

Hello,

I have the following integral:

$$\int_0^{\gamma}(x)^{a}\left(x^2+x\right)^{b}\mbox{exp}(cx) K_{(2b)}\left(2d\sqrt{x^2+x}\right)\,dx$$

but I didn't find any equivalent integral in the table of integrals. Now, how write this integral in closed-form expression using known functions?

Note: This is not a homework, but general treatment of an integral.

Regards

 

[CRGreathouse]

What is $K_{(2B)}$?

W|A has nothing.

 

[AUMathTutor]

W|A? This is the math forum, not the joke forum.

On a more serious note, we're going to need more information. For instance, ...
what the heck is the K thing?
why should this have a closed-form expression?

 

[EngWiPy]

W|A? This is the math forum, not the joke forum.

On a more serious note, we're going to need more information. For instance, ...
what the heck is the K thing?
why should this have a closed-form expression?

$$K_{(2b)}(.)$$ is the modified Bessel function of the second kind and $$(2b)^{th}$$ order. I need it in closed form because I want to continue my derivation based on this integral.

 

[HallsofIvy]

Then the question becomes "Do you have reason to believe that such a closed form exists?" In a very precise sense "almost all" integrable functions do not have an integral that can be written a closed form in terms of known functions.

 

[EngWiPy]

Then the question becomes "Do you have reason to believe that such a closed form exists?" In a very precise sense "almost all" integrable functions do not have an integral that can be written a closed form in terms of known functions.

Excluding the "closed form expression", is this integral is solvable? i.e.: can be found indirectly in the table of integrals? It needs an expert in math, who knows many integrals to connect them in according, I guess.

 

[CRGreathouse]

You can certainly calculate it numerically. But you probably won't be able to find it in a table of integrals because there probably isn't a closed-form solution. Tables of integrals only have expressions with closed-form solutions.

 

[EngWiPy]

You can certainly calculate it numerically. But you probably won't be able to find it in a table of integrals because there probably isn't a closed-form solution. Tables of integrals only have expressions with closed-form solutions.

Ok, fine. How can I calculate this integral manually? what is the first step to do?

 

[EngWiPy]

So, we simply can't solve the integral?

 

[uart]

So, we simply can't solve the integral?

No, you can pick some numbers for a,b,c,d and gamma and then solve it numerically using something like Simpson’s rule. The problem is that, in it's current form, you've got five dimensions to play around with, and that's means a lot of individual data points that you'll have to numerically evaluate if you want to get some kind of overview of what that function is doing.

The more parameters that you can fix the easier it will be. If for example there was a way that you could sensibly pick numerical values for four of those five parameters then it would be easy get the data points to plot the integral versus the remaining free parameter.

 

[EnumaElish]

You may be able to write the integral of K as a series, at, say, x=0:

For b =1, ∫K(2)[x] dx = x K(2)[0] + x^2 K(2)'[0]/2 + ...

 

[EngWiPy]

You may be able to write the integral of K as a series, at, say, x=0:

For b =1, ∫K(2)[x] dx = x K(2)[0] + x^2 K(2)'[0]/2 + ...

Yes, but the modified bessel function is multiplied by other functions in the integral, so, we can not separate it.

 

[EngWiPy]

No, you can pick some numbers for a,b,c,d and gamma and then solve it numerically using something like Simpson’s rule. The problem is that, in it's current form, you've got five dimensions to play around with, and that's means a lot of individual data points that you'll have to numerically evaluate if you want to get some kind of overview of what that function is doing.

The more parameters that you can fix the easier it will be. If for example there was a way that you could sensibly pick numerical values for four of those five parameters then it would be easy get the data points to plot the integral versus the remaining free parameter.

All of these parameters are constants (a, b, c, and d). I neither want to plot the integral, nor evaluate it numerically, but I want in a some way to find a closed-form expression for this integral, if it exists.

 

[g_edgar]

Maple does not compute a closed for even for

$$\int K_0\big(2\sqrt{x^2+x}\big)\,dx$$

so your more elaborate one is even less likely