Difficulty with evaluating an integral

In summary, the conversation discusses a problem with evaluating the anti-derivative of a differential binomial expression. The person has attempted to solve it using standard integration techniques but it becomes increasingly complicated. They have also tried to manipulate it into a perfect square but that failed. They then tried using Wolfram Alpha but the answer involves a hypergeometric function, indicating that the anti-derivative cannot be written in terms of elementary functions. The conversation also mentions the concept of differential binomials and how they can affect integration.
  • #1
throneoo
126
2

Homework Statement



How to evaluate the anti-derivative of [x+(7x^2)]^(1/3) ?

2. The attempt at a solution

I've attempted to do so via standard integration techniques , i.e. substitution and integration by parts , but the problem simply becomes increasingly complicated . I've even tried to manipulate it into a perfect square in hopes of getting an easier expression but that failed too.

My substitution : u=(x+7x^2) ; du=(1+14x)dx and x=(+/-(1+28u)^(0.5)-1)/14
making the intergrand u^(1/3)(1+28u)^-0.5

with by parts things get so complicated it terrified me.

Frustrated , I tried wolfram alpha ,only to get sth like this :
http://www4c.wolframalpha.com/Calculate/MSP/MSP4182022fe0gi4018be9000021e9375hf3bi1a91?MSPStoreType=image/gif&s=36&w=569.&h=56. [Broken]

I don't even know what a hypergeometric function is

Can anyone help me ?
 
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  • #2
If Wolfram alpha is giving the answer in terms of a "hypergeometric" function, I would take that as an indication that the anti-derivative cannot be written in terms of "elementary" functions. Where did you get that integral and what reason do you have to believe that it can be written in terms of elementary functions?
 
  • #3
It was from the book Feynman's Tips on Physics , where the author mentions integration as a prerequisite for doing physics.

the 5 examples of integrand he gives are (1+6t) , (4t^2+2t^3) , (1+2t)^3 , (1+5t)^0.5 and [t+(7t^2)]^(1/3).
the first 4 examples' anti-derivatives are given in terms of "elementary functions" , while the last one is left with some question marks . so I thought the last one was meant as an exercise and decided to give it a try , assuming it's as simple as the other examples without any particular reasons.

Edit: upon a closer look into the text , Feynman says it isn't necessary to be able to integrate simple expressions and the last one is not possible to integrate in an easy fashion , implying it's not really as simple as it looks . lol
 
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  • #4
This is a differential binomial.It's an expression of the type [itex]x^{m}(a+bx^{n})^{p}dx[/itex]
In your case that becomes [itex]x^{1/3}(1+7x)^{1/3}dx[/itex]
If [itex]\frac{m+1}{n}[/itex] is an integer you can make the substitution [itex]u = ax^{n}+b[/itex]
If [itex]\frac{m+1}{n}+p[/itex] is an integer you can make the substitution [itex]u = a+bx^{-n}[/itex]
In all other cases this integral cannot be expressed by elementary functions
 
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  • #5
exo said:
This is a differential binomial.It's an expression of the type [itex]x^{m}(a+bx^{n})^{p}[/itex]
In your case that becomes [itex]x^{1/3}(1+7x)^{1/3}[/itex]
If [itex]\frac{m+1}{n}[/itex] is an integer you can make the substitution [itex]u = ax^{n}+b[/itex]
If [itex]\frac{m+1}{n}+p[/itex] is an integer you can make the substitution [itex]u = a+bx^{-n}[/itex]
In all other cases this integral cannot be expressed by elementary functions

wow..differential binomials must be quite some significant objects to be studied so precisely...
Thanks for the help . I guess I can't go anywhere further in this problem with my current knowledge
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a continuous function over a given interval.

2. Why is it difficult to evaluate an integral?

Evaluating an integral can be difficult because it involves finding an anti-derivative or the original function that would produce the given derivative. This process can be complex and may require advanced mathematical techniques.

3. What are some strategies for evaluating integrals?

Some strategies for evaluating integrals include using substitution, integration by parts, and partial fractions. It is also helpful to have a strong understanding of basic integration rules and techniques.

4. How can I check if my answer to an integral is correct?

One way to check the correctness of an integral is to take the derivative of the result and see if it equals the original function. Another method is to use a graphing calculator or software to plot the original function and the integral to see if they match.

5. What are some real-world applications of integrals?

Integrals are used in many real-world applications, such as calculating areas and volumes in engineering and physics, determining rates of change in economics and biology, and finding probabilities in statistics and finance.

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