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Evaluate the indefinite integral as an infinite series.

$ \int \sqrt {1 + x^3} dx $

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$C+\Sigma_{n=0}^{\infty} \left( \begin{array}{c}{1 / 2} \\ {n}\end{array}\right) \frac{x^{3 n+1}}{3 n+1}$

01:02

Wen Zheng

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

University of Nottingham

Idaho State University

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

01:38

Evaluate the indefinite in…

03:29

Evaluate the integral.

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so in this probably want to calculate an indefinite integral, um, and we'll be using, um, the binomial serum to help us solve this. So we have is the indefinite integral of the square root of one plus X cubed the X. So with that, we want to use the binomial theorem, which essentially states one plus X to the K is equal to the summation from an equal zero to infinity of Okay, um, for the lack of having being able to write this will just UK over x x two then this is not division. Just to be clear, it's just k over the X. Then we know that X is going to be X cubed, and K is going to be one half. So what will end up getting as a result is one plus one plus X cubed, being equal to one plus x cubed half. And this is the square root of one plus X cubed to be clear. Yeah, we know that these two are equal to each other. So the way that we're going to write this in terms of the summation is that it's equal to the sum from zero to infinity of Okay, one half over and again. That's just that's not supposed to be a fraction times X cubed. And with this in mind, um, what we have is X cubed to the end when we are now, take the integral of this the ax. We end up getting that. This is the summation from end to infinity of one half over and, um, of X to the three and plus one over three. The an plus one. I see. And that will be our final answer. Um, just the main thing to look out for him be careful of is to make sure that there's no fraction bar right here. Um, that's just one half on the top and and on the bottom.

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