Integrating irrational functions

[Marin]

Hello everyone!

I was wondering.. if you could help me calculate some integrals:

It's not for Homework or something, just my curiosity:

$$\displaystyle{\int}\sqrt[3]{x^2-1} dx$$

What would you suggest? I tried substitution, thou it seems to me useless.

Are these integrals common in tests and exams? I mean, they seem to me just pure mathematics, i.e. - no application, and I'm not sure, if that's not too much to ask from a student..

Thanks in Advance, Marin

 

[HallsofIvy]

If it weren't for that "3", which I almost missed, I would say use a trig substitution. Since that won't work, I have to ask if you have any reason to think it does have an elementary anti-derivative? You must understand that "almost all" elementary integrands do not have an elementary anti-derivative. ("Elementary" here meaning made up of combinations of polynomials, roots, trig functions, exponentials, logarithms.)

 

[Marin]

hmmmm, didn't think of that

Maybe it doesn't. That means, we have to integrate using power series, to get the series of the antiderivative, doesn't it?

 

[tiny-tim]

Hello everyone!

I was wondering.. if you could help me calculate some integrals:

It's not for Homework or something, just my curiosity:

$$\displaystyle{\int}\sqrt[3]{x^2-1} dx$$

Hello Marin!

hmm… I get ∫(sinhu)^5/3 du, which (like HallsofIvy ) I don't think has a "elementary" anti-derivative.

Are these integrals common in tests and exams? I mean, they seem to me just pure mathematics, i.e. - no application, and I'm not sure, if that's not too much to ask from a student.

If I were you, I'd just do the set examples, and similar ones.

Your course has been specially designed to make the best use of your time.

There are ways of integrating such functions (well, there must be, mustn't there? ), but they're in a different ball-park, and you won't be able to guess them from what you already know.

Your professors will induct you into that ball-park when and if the time is appropriate …

trust your professors, and follow the course!

 

[Marin]

Well I guess I have no other choice but wait and follow the profs :)

Sometimes I tend to search for solutions of examples not to my level.. but what can I do for it - they just don't leave me in peace.., despite I think of self-teaching as not very appropriate

 

[omenka]

The integral you submitted has no elementary anti-derivative, ever heard of Chebychev criteria for irrational integrals? If not ask me, "the integrator guru", Omereada Nke Mbu

 

[g_edgar]

This definite integral is a Beta function...

$$\int _{0}^{1}\!\sqrt [3]{1-{x}^{2}}{dx}=2/15\,{\frac {{\pi }^{2}{2}^{2 /3}}{ \left( \Gamma \left( 2/3 \right) \right) ^{3}}}$$

 

[daudaudaudau]

What did people do back in the day with no computers, when they had to determine if a function had an antiderivative? Is there a way of doing it? I have heard about the Risch algorithm, but that doesn't work in all cases.

 

[Tac-Tics]

What did people do back in the day with no computers, when they had to determine if a function had an antiderivative? Is there a way of doing it? I have heard about the Risch algorithm, but that doesn't work in all cases.

"Back in the day" (as well as in modern times), it's usually a matter of seeing the function by a more general description of it.

A real function is integrable on an interval [a, b] whenever it's bounded and continuous almost-everywhere on [a, b]. This theorem is one of the first things you can prove about the Riemann integral. And it's not terribly difficult to believe (or even prove) that "nice" algebraic functions like your example here fit those conditions, and are integrable.

 

[Дьявол]

He could try to solve it.
$$\displaystyle{\int}\sqrt[3]{x^2-1} dx$$

$$t^3=x^2-1$$

$$3t^2dt=2xdx$$

$$\frac{3}{2}\int{\frac{t^3}{\pm \sqrt{t^3+1}}dt}$$

Can you solve it ?

Regards.

 

[daudaudaudau]

"Back in the day" (as well as in modern times), it's usually a matter of seeing the function by a more general description of it.

A real function is integrable on an interval [a, b] whenever it's bounded and continuous almost-everywhere on [a, b]. This theorem is one of the first things you can prove about the Riemann integral. And it's not terribly difficult to believe (or even prove) that "nice" algebraic functions like your example here fit those conditions, and are integrable.

Sorry, I meant to ask: How do you know if a function has an antiderivative which is expressible in closed form in terms of elementary functions? Because otherwise you could spend a lot of time trying to integrate somthing which really has no elementary antiderivative!

 

[Ben Niehoff]

One can prove that there is no algorithm which can determine whether a general function has an elementary antiderivative. This is one of those things that must be discovered through trial and error...and many elementary integrals turn out to be non-obvious, such as

$$\int \sec x \; dx = \ln \left| \sec x + \tan x \right| + C$$

However, algorithms do exist for specific classes of functions, to determine whether they have elementary antiderivatives.

Note, this is in stark contrast to finding derivatives; there exists a step-by-step process to find the derivative of any function quit easily!

 

[Дьявол]

Have anybody sow my solution?

Here is the next step (continuing from post #10):

$$= \frac{3}{2}\int{\frac{t^3+1-1}{\pm \sqrt{t^3+1}}dt}=\frac{3}{2}(\int{\frac{\sqrt{(t^3+1)^2}}{ \sqrt{t^3+1}}dt}-\int{\frac{1}{\sqrt{t^3+1}}dt})=$$

$$=\frac{3}{2}(\int{\sqrt{t^3+1}}dt}-\int{\frac{1}{\sqrt{t^3+1}}dt})$$

Regards.

 

[omenka]

ever heard of Chebychev criteria for integrating irrational expressions? That tells you whether or not you can perform the integration

 

[omenka]

The Chebychev's criteria states conditions under which some irrational functions can be integrated. There are also several books that give excellent account on integral calculus; starting from the book by Abramowitz and Stegun (Handbook of Mathematical Functions) to a book written in German with accompanying English translation, Summen, Produkt und Integrale by Ryshik and Gradstein. For more information on harder integrals send me a mail (<< e-mail address deleted by Mentor >>). I am also on facebook under Omereada H. Omereada, my grandfather's title.

Henri Onuigbo

 

[HallsofIvy]

What did people do back in the day with no computers, when they had to determine if a function had an antiderivative? Is there a way of doing it? I have heard about the Risch algorithm, but that doesn't work in all cases.

Essentially people did integrations with hand cranked adding machines (I used to have one!) and then put the results in tables that other people could get the values out of. I remember seeing, in the University of Florida library, a 12 foot shelf of books titled "Tables of Elliptic Integrals".