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I am working on a final project for Calculus 2 and want to make sure that I am on the right track and looking for some constructive criticism on how to improve my work .. Here is the problem

Part I: Integration

Over the course of Calculus I and Calculus II we have learned many methods for integrating functions. These range from using a basic formula and substitution to more complex rules like by parts and trig substitution. In this part of your project do the following:

1) Create a flowchart that will allow anyone given a random problem to decide which of all the rules, processes, and/or procedures they should use to solve it. You must include all of the following in your flow chart:

a. Basic Formulas

b. Substitution

c. Numerical Integration

d. Integration by Parts

e. Integration using trigonometric rules

f. Integration using trigonometric substitution

g. Integration tables

2) Explain how to use your chart to decide the best way to solve the following problem.

Note: You do not need to solve the problem.

Here is my work:

In Calculus there are many problems that have to be solved using a particular method of approach. With Integration there are eight methods in particular that we talk about. The eight methods are: basic formulas, basic substitution, numerical integration, integration by parts, integration using trigonometric rules, integration using trigonometric substitution, inverse trigonometric formula, and the integration tables. When given an integration problem it is best to first analyze the problem and break it into pieces and see if there is a way to better work with it.

For example given the problem $\displaystyle \int \dfrac{6x}{(x^2+9)^2}dx$ we can simply change the formula to look like $$\displaystyle \int \dfrac{6x}{(x^2+9)^2}dx=3\int \frac{du}{u^2}$$ making it more obvious which way we should choose to integrate the problem.

By transforming the problem we are able to use basic substitution, which is the first item on the flow chart. Basic substitution, the table of integrals, and the basic formulas are the easiest methods available to try and complete the methods at hand. If we were unable to use the first item on the flow chart we would have to follow the no path to the next method, so on and so on until we found a method that would work. When first given the problem the easiest way to go about using the table is to convert your problem from numbers to variables so that you can get a better idea of what your problem actually looks like.

I have a flowchart that I made up in Visio will be attached along with this written portion.

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3  
@Zetta That sort of correction changes the content of the post. I don't think it's right to do it. It is my opinion that you should alert the OP for the mistake and not edit it correctly. –  Git Gud Jun 23 '13 at 16:26
1  
@GitGud I see edits correcting math mistakes as no different than spelling/grammar corrections. Especially when the correctness of a particular expression is not the subject of the question. Besides, my edit was just a suggestion, and it was accepted. –  Ataraxia Jun 23 '13 at 16:46
    
I thought that with integrals what you do to the inside you have to invert and do to the outside. On this problem since we are multiplying inside by 3 we would multiply outside by 1/3? –  Tonya Jun 23 '13 at 19:52
    
@Tonya We're dividing the inside by $3$ not multiplying. There's already a $6$ inside the integral, so in order to get rid of it we have to divide by $3$. A factor of $3$ therefore goes on the outside to reflect that. –  Ataraxia Jun 23 '13 at 22:55
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1 Answer 1

First it is good to be aware that the work will depend on how precise or effective you want the flow chart to be. There are algorithms (and therefore are flow chart) that will give you the answer of what to do for every elementary function (if it has an elementary function as primitive). You can begin to read about it here. (http://en.wikipedia.org/wiki/Risch_algorithm)

Another thing is that the flow chart can be given so that it ramifies in many different ways. For example, the result of Liouville that is mentioned in that same Wikipedia article shows, that all of these problems can, in a sense, be solved by substitution alone (plus algebraic manipulations).

You can fill a good chunk of the flow chart with the method for integrating all rational functions. Then add classes of functions that can be reduced to integration of a rational function by some substitution. Again, in principle, this covers all the cases, but you probably want to keep the substitutions to be as simple as possible. Reduction to a rational function is not always the fastest way to get some primitives.

Classes that are nice to include:

  1. Rational functions.
  2. Rational functions times the logarithm of a rational function.
  3. Rational functions times the sin (cos) of a rational function.
  4. Rational functions times the exponential of a rational function.
  5. Rational function times the $\arctan$ of a rational function.
  6. Rational function composed with $\sin(x)$ and $\cos(x)$, i.e. of the form $R(\sin(x),\cos(x))$ with $R(x,y)$ rational.
  7. Rational function composed with the square root of a linear polynomial. $R(\sqrt{ax+b},x)$.
  8. Rational function composed with the square root of a quadratic polynomial. $R(\sqrt{ax^2+bx+c},x)$.
  9. Rational function composed with the square root of a fractional linear transformation. $R(\sqrt{\frac{ax+b}{cx+d}},x)$.

I am not sure now if this covers all functions treated in textbooks, but it should cover a good part of them.

Maybe in $3$ and $4$ it is better to only look at linear functions inside the $\sin$, $\cos$, and the exponential to avoid the cases in which there is no elementary primitive. I am not sure now in $2$ if it is going to happen.

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