Evaluating Integral and Series: Find Solution to ∫√(x³ +1)dx

[merced]

1. Homework Statement Evaluate the indefinite integral as an infinite series.
$$\int{\sqrt{x^3 +1}}$$

Homework Equations

Taylor Series: f(x) = $$\Sigma_{n=0}^{inf} \frac{\f^{n}(a)}{n!} (x-a)^n$$

Maclaurin Series --> taylor series with a = 0

The Attempt at a Solution

I have no idea how to evaluate this integral. I can't use u substitution. From the chapter, I have seen no general formula for such an integral.
Please give a hint.

 

[Hurkyl]

Firstly, I'm going to assume you meant to write $\int \sqrt{x^3 + 1} \, dx$.

Please give a hint.

Read the problem!

Evaluate the indefinite integral as an infinite series.​

This suggests infinite series are going to be involved. Is there anything here you can write as an infinite series?

 

[merced]

Yeah, I know that I'm supposed to use an infinite series...but my book only gives examples that are similar to e^x, sin x, cos x, $$tan^{-1}x$$, and 1/(1-x)...so I can't really think of what to do.

$$\int{\sqrt{x^3 +1}}$$ is not similar to any of those, except maybe the integral of $$tan^{-1}x$$.

 

[Hurkyl]

Well, you've learned how to compute infinite series, haven't you? You could try that.


P.S. Your book doesn't have infinite series expressions for things like $\sqrt{x+1}$?

 

[prace]

Could you solve by first expanding the integrand using binomial expansion and then integrate each individual part?

 

[merced]

Umm...I don't think so. My professor did not cover this section in the detail I would have liked.
I didn't quite understand everything in the section.

Please tell me how to compute infinite series! :)

 

[merced]

I haven't gotten to binomial expansions yet.

 

[prace]

Oh I see, well, just in case you want to know, the binomial expansion series says you can expand a binomial according to the following formula:

$$1+kx+\frac{k(k-1)}{2!}x^2+\frac{k(k-1)(k-2)}{3!}x^3$$ and so on, where k is your exponent.

In this integral, $$\int{\sqrt{x^3 +1}}$$ your function is raised to the (1/2) power so k = (1/2). Then you just simply plug in for x and k in the formula for a couple of terms and integrate each term individually.

I worked out your integral on my TI-89 and evaluated from 0 to 1 and got the answer 1.111448. When I worked it with the binomial expansion method (3 terms) I got 1.149. So you can see the values are pretty close. If I worked out more terms, my answer would have been any closer.

I know you want to use a different technique, but just in case you can't find it through your method, I just wanted to give you an alternative method to use. Hope it helps and I wish I could help you with the other way, but I am not too sure about that way.

 

[Hurkyl]

Please tell me how to compute infinite series! :)

You already know how!

As you've said (correcting the typos):

$$f(x) = \sum_{n=0}^{+\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$$

at least... you know how to compute the first several terms of the infinite series. (And hopefuly, you can then figure out what the general term is)

 

[dextercioby]

Oh I see, well, just in case you want to know, the binomial expansion series says you can expand a binomial according to the following formula:

$$1+kx+\frac{k(k-1)}{2!}x^2+\frac{k(k-1)(k-2)}{3!}x^3$$ and so on, where k is your exponent.

In this integral, $$\int{\sqrt{x^3 +1}}$$ your function is raised to the (1/2) power so k = (1/2). Then you just simply plug in for x and k in the formula for a couple of terms and integrate each term individually.

I worked out your integral on my TI-89 and evaluated from 0 to 1 and got the answer 1.111448. When I worked it with the binomial expansion method (3 terms) I got 1.149. So you can see the values are pretty close. If I worked out more terms, my answer would have been any closer.

I know you want to use a different technique, but just in case you can't find it through your method, I just wanted to give you an alternative method to use. Hope it helps and I wish I could help you with the other way, but I am not too sure about that way.

Incidentally, the binomial formula you wrote is valid only when k is an integer. So advising him to plug k=1/2 is nonsensical.

Daniel.

 

[Hurkyl]

Actually, the name "binomial formula" is often applied to the Taylor series for (1+x)^k, even when k is not an integer. You can even write it with binomial coefficients, if you take the generalized definition.