Euler substitution (Good sources)

In summary, these three substitutions are used to evaluate integrals. They are called euler substitutions because the Swiss mathematician, Leonhard Euler, first described them. The first substitution is for when integrating with u, which has the derivative of u multiplied elsewhere in the integrand. The second substitution is for when integrating with v, which has the derivative of v multiplied by the derivative of x. The third substitution is for when integrating with x, which has the derivative of x multiplied by the derivative of y. These substitutions are useful when the square root of the integral cannot be found using other methods, such as trigonometric substitution.
  • #1
Nebuchadnezza
79
2
Anyone know any good sources to read about Euler substitutions from?

Both online and books would be suffice. Also videos, if there are any.

All help would be greatly appreciated =)
 
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  • #2
Shouldn't it be in any integral and differential calculus books?
 
  • #3
Well it is not in my book =(

And ofcourse different books explain things in different ways. Some good and some horrendous. So I was wondering if anyone knew about a book having a clear, good explenation of it.
 
  • #4
Are you meaning integrals and derivatives with e^x, e^u, ln x & ln u ?

A solid grasp of the substitution guidelines is a great starting point.

Are you looking principles like, when integrating,
Choose the most complex u, which has the derivative of u multiplied elsewhere in the integrand.
?

I don't or rarely see those principles written in textbooks. Usually, the professor or a tutor is the best source for guidelines. Most colleges have free tutoring clinics available with varying instructional quality. Several in my area will tutor students from other colleges.
 
  • #5
I was talking about three substitutions, I thought they were called euler substitutions. But I had a hard time finding them left.

[tex] \sqrt{ax^2+bx+c}=t-\sqrt{a}x [/tex]

[tex] \sqrt{ax^2+bx+c}=xt + \sqrt{c} [/tex]

[tex] \sqrt{ax^2+bx+c}=x(t-c)[/tex]'


Sorry about my english, nt sure what these are called.
 
  • #6
Again give us a context.
Are these for evaluating integrals? For differential equations? Something else like cubic and higher degree equations?
 
  • #7
Yeah, they are for evaluating integrals. Called euler substitutions, or atleast I think they are calle thath.

I sort of want to know when these are useful, when to preffer one over the other one. When to use this over for example trigonometric substitution. Also wondering if there are any integrals that "must" be solved by this method

Just looking for books about this =)
 
  • #8
I think what you may be looking for is Abel substitution. It is basically t = sqrt(ax^2+bx+c). After that a simple linear transformation will give your examples.
 
  • #9
I doubt you'll find an entire book on it. One might find a section using it or an entire chapter using it.
I google Able substitution which led me to Abel transforms. Not much success there.
Can you give me some example integrals you're trying to evaluate?
I can probably figure out application methods with more details.

Neither
http://www.wolframalpha.com/input/?i=integral+1%2F+root+%28ax^2%2Bbx%2Bc%29+dx
nor
http://www.wolframalpha.com/input/?i=integral+root+%28ax^2%2Bbx%2Bc%29+dx
are known to WolframAlpha, the most extensive, free computer integrator that exists. So I need at least a few examples to help you.
 
  • #10
Correction to previous post:
http://www.wolframalpha.com/input/?i=integral+%28ax^2%2Bbx%2Bc%29+^.5dx
and
http://www.wolframalpha.com/input/?i=integral+1%2F+%28ax^2%2Bbx%2Bc%29+^.5dx

both evaluate. The show steps feature on the first link shows the substitution and uses one of yours. Notice how WolframAlpha, abbreviated W|A, first completes the square, then does the substitution?
So we've got a way to see how it works in practice. Perhaps ask WolframAlpha enough integrals and you figure out the how and why?

As to when to use these vs. when to use trig. substitution, trig. substitution on [itex]\sqrt{trinomial}[/itex] is probably even uglier than the method W|A shows. So wouldn't it make sense to use these for [itex]\sqrt{trinomial}[/itex], but trig. substitution for forms similar to [itex]\sqrt{1\pm x^2}[/itex]?

I've reached the limit of how I can help. God bless you & bye.
 
  • #11
One more thought occurred to me. The choice of which of the three t expressions to substitute will be influenced by what makes the square root real? Unless, you've learned or are learning calculus of complex numbers.
 

1. What is Euler substitution?

Euler substitution is a mathematical technique used to simplify integrals involving radical expressions, specifically those containing square roots or cube roots. It involves making a substitution of variables in order to transform the integral into a more manageable form.

2. When is Euler substitution used?

Euler substitution is typically used when dealing with integrals that involve expressions of the form √(a^2 - x^2), √(x^2 - a^2), or √(x^2 + a^2). It is also useful for integrals involving trigonometric functions.

3. How do you perform Euler substitution?

To perform Euler substitution, you first need to identify the radical expression in the integral. Then, make a substitution of variables, usually using trigonometric functions, to eliminate the radical. This will transform the integral into a more manageable form, which can then be solved using traditional integration techniques.

4. Are there any limitations to using Euler substitution?

While Euler substitution can be a useful technique for simplifying certain integrals, it is not always applicable. It may not work for integrals with more complex radical expressions or those involving higher powers of radicals. Additionally, it may not always lead to a solution, so other integration techniques may need to be used.

5. Where can I find good sources for learning about Euler substitution?

There are many online resources available for learning about Euler substitution, including math websites, video tutorials, and online textbooks. It may also be helpful to consult with a math teacher or tutor for additional guidance and practice problems.

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