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

• merced
In summary, the conversation is about evaluating an indefinite integral as an infinite series. The conversation discusses using Taylor series and Maclaurin series to evaluate the integral, as well as the binomial expansion method. The final suggestion is to use the Taylor series for (1+x)^k, even when k is not an integer, as an alternative method to evaluate the integral.
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.

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

merced said:

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?

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 $$tan^{-1}x$$.

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 $\sqrt{x+1}$?

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

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! :)

I haven't gotten to binomial expansions yet.

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.

Please tell me how to compute infinite series! :)

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)

prace said:
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.

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.

## 1. How do I solve an integral with a square root?

To solve an integral with a square root, you can use substitution or integration by parts. In this case, you can let u = x³ + 1 and then use the power rule for integration to solve the integral.

## 2. Can I use a calculator to solve this integral?

Yes, you can use a calculator to solve this integral. However, it is important to understand the steps involved in solving an integral by hand in order to properly use a calculator and interpret the results.

## 3. What is the difference between an integral and a series?

An integral is a mathematical concept used to find the area under a curve, while a series is a sum of infinitely many terms. In this case, we are evaluating an integral, not a series.

## 4. How do I know if my solution to an integral is correct?

To check if your solution to an integral is correct, you can take the derivative of your answer and see if it matches the original function. You can also use online integral calculators or ask a math tutor for assistance.

## 5. Are there any special techniques for solving integrals with square roots?

Yes, there are special techniques for solving integrals with square roots, such as using trigonometric substitution or completing the square. It is important to practice these techniques and understand when to use them in order to master solving integrals with square roots.

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