Integrating (e^4x)/x

You could use integration by parts. If it helps, rewrite it as:
x^-1*e^4x

 

I don't think that integration by parts is going to be too helpful here. This is definitely not a standard textbook exercise in basic integration.

 

By definition, it is -Ei(1,-4x). Where Ei is the exponential integral.

Hope this helps.

;0

 

Using a substitution, this integral can be simplified to [tex] \int \frac{e^u}{u} du [/tex]
No matter how hard you try, you can never succeed in integrating this integral. It cannot be defined by any known functions, much like [tex] \int \sin(x^2) dx [/tex] and [tex] \int \frac{sinx}{x} dx [/tex].

 

mlleRosie is correct. There is no current known method for integrating this type of equation.

Simply put, im sure your familiar with the chain rule, power rule, Lhoptials rule, etc... The method to solve this is similar to those... it just hasnt been discovered yet. In theory there is a number formula (like Log, sine, tan, cos, etc) that hasnt been figured out and that will be used in the new formula. Cool huh?

Well go get yourself a nobel prize and invent the shansalman's rule for integrating this!

 

Some others I just came up with are

Integrate: square root(1+x^3) dx

integrate: e^x^2 dx

How do you guys do that latex stuff?

 

fixed.

I don't think they'd give a nobel prize for that, unless it had a very significant use in physics, ecomonics, chemistry, or biology.

 

Im sure your right. It would be amazing nevertheless.

Just think, there is something that exists out there that will probably be taught at the high school level once its discovered. Its something simple yet nobody has figured it out yet. (This whole paragraph is obviously a maybe).

Just its really exciting isnt it ?!?!?!

Maybe this new function will relate some of the major theories out there (e = mc^2 and some others) and we can finally prove the grand unified field theory (the everything theory). Then im sure it will get a nobel and maybe we could travel to the stars! OMG im so excited now! Its like wondering "what if" if you won the lottery.

 

Found another one:

Integrate sin(x^2)

 

The function has been discovered already. ZioX mentions it in the third reply to this thread. It is the Exponential Integral function, one of a family of functions which has been studied by many a mathematician.

 

EUREKA!!! we are all going to be rich! :-) that is too cool. im behind in my reading it looks like!

 

I haven't been to that page but I assure all here that the Exponential Integral Function is just a definite integral, not an anti-derivative. And that is no amazing achievement, I can do the same for any function, watch.

[tex]\int f(x) dx = F(x) + C, \frac{dF(x)}{dx} = f(x)[/tex].

Now if i wanted, I could study the properties of a particular F(x), as they did for the Exponential Integral Function.