I've been trying to integrate this function by substitution, and it doesn't seem to be getting me to the correct place. I'm not sure I fully understand how to use substitution.

 

[math]

y = \int\sqrt{x^3 - 1} dx

[/math]

 

I've only ever dealt with substitutions where you will and up with

[math]

du = a dx

[/math]

where a is a constant, but if I make the substitution

[math]

u = x^3 - 1

[/math]

then I end up with

[math]

du = 3x^2 dx

[/math]

 

And you can't just slap 3x² back into the the integrand. Can someone integrate this and tell me what you need to do?

This function is not integrable in the elementary functions. In other words, you can use all the tricks you know (and all that you don't know) and you will not find the integral.

You need to look up the elliptic integral of the first kind in order to solve this. Though as DH has said, you cannot get the integral as a closed expression involving elementary functions.

You need to look up the elliptic integral of the first kind in order to solve this.

I thought about telling the OP about elliptical integrals, but then rejected that thought since theOP is "not sure I fully understand how to use substitution."

Well, the aim of this post was to learn something about integration, I now realise why this function can't be integrated to another function, because this function clearly isn't a product, and no substitution will leave you with du as a scaled form of dx. Sorry for the basic terminology, but I understand why it can't be integrated now!

 

Now, although I'm perfectly aware I'm probably getting ahead of myself, what is this "Elliptic Integral" supposed to achieve?

Well, the aim of this post was to learn something about integration, I now realise why this function can't be integrated to another function, because this function clearly isn't a product, and no substitution will leave you with du as a scaled form of dx. Sorry for the basic terminology, but I understand why it can't be integrated now!

The problem is not that this integral doesn't exist. It most certainly does exist. The problem is that this integral can't be expressed in terms of the elementary functions. All the neat tricks that you have learned are aimed at solving those integrals than can be expressed in terms of elementary functions.

 

Most integrals cannot be expressed that way. For example, students regularly ask how to integrate [imath]\exp(x^2)[/imath] or [imath]\exp(-x^2)[/imath]. Play around with u-substitutions, integration by parts, etc., and no matter what you do you will either get an intractable mess or just get right back to square one.

 

That doesn't mean the integral doesn't exist. It does exist. It turns out that [imath]f(x) = \frac{\,2}{\surd\pi}\int_0^x \exp(-t^2)\,dt[/imath] is a very important function. It appears so often, and in so many different settings, that it has been given a special name (the error function) and a special symbol [imath]\textrm{erf}(x)[/imath]. The error function is just one of many integrals that can't be expressed in terms of elementary functions and that rear their ugly heads over and over again. These "special functions" will be the subject of multiple advanced math classes should you go that far in math.

 

Now, although I'm perfectly aware I'm probably getting ahead of myself, what is this "Elliptic Integral" supposed to achieve?

Amongst other things, the arc length along a segment of an ellipse. Elliptical integrals are one of those things that keep coming up again, and hence they are given a special name.

Edited April 26, 201213 yr by D H

So, when you say that many integrals cannot be expressed in terms of elementary functions, are you suggesting that there are other ways of expressing integrals? Or are many functions just such that they cannot be "expressed" at all, in any other way than just "the integral of another function"? Sorry, calculus just interests me so much!

 

Taylor series seem to be coming up alot in expressing these complicated integrals! hmmmm, thankyou anyway, this will give me lots to mull over

Naming the solutions of particular integrals was a speciality of classical mathematical physics. The rather general term used is "special functions", though there is no tight definition here. Special functions are usually either solutions to differential equations or integrals of elementary functions.