Solving Tough Integral with Log - Need Help

In summary, the conversation discusses the difficulty of computing an integral involving a square root and a logarithm. The variables involved are E1, m, and mB, and the integration limits are from m to mB/2. The original poster is looking for an algebraic solution and has tried using substitution methods, but nothing seems to simplify the integral. They mention using Mathematica and trying to give it assumptions for the variables, but are still struggling with finding a solution.
  • #1
Hepth
Gold Member
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I posted this in the mathematical/computation software forum but maybe there's an algebraic trick I don't know to help me solve this.

I need to compute an integral of:
[tex]\sqrt{\text{E1}^2-m^2} Log\left( \frac{m^2-m_B E1-m_B\sqrt{E1^2-m^2}}{m^2-m_B E1+m_B \sqrt{E1^2-m^2}}\right)[/tex]

over the region {E1,m,mB/2}

the limits of integration are well defined in the equation, though the function is asymptotic near the upper bound. But it still has a value there.

I can't seem to integrate this. Mathematica just spits the input back out. I know its possible because I can numerically integrate it for what I want and it gives a reasonable answer. Unfortunately I REALLY want an algebraic solution...

Does anyone have any ideas? An integral of a sqrt times a log of a function of the variable. I've tried some substitution methods, nothing seems to simplify.

Thanks for the help!


EDIT:I think I got something by letting U= Log[everything], it can be algebraic, but its sooo long... any other choices?
 
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  • #2
Hepth said:
I posted this in the mathematical/computation software forum but maybe there's an algebraic trick I don't know to help me solve this.

I need to compute an integral of:
[tex]\sqrt{\text{E1}^2-m^2} Log\left( \frac{m^2-m_B E1-m_B\sqrt{E1^2-m^2}}{m^2-m_B E1+m_B \sqrt{E1^2-m^2}}\right)[/tex]

over the region {E1,m,mB/2}
What is the variable here? Why are there three integration limits {E1,m,mB/2} instead of two?
 
  • #3
variable is E1. From m to mB/2

I used mathematicas way of saying that sorry.
 
  • #4
Hepth said:
EDIT:I think I got something by letting U= Log[everything], it can be algebraic, but its sooo long... any other choices?

Well, you can also use Mathematica do perform these steps and simplify the end result...
 
  • #5
It looks like you might be able to rewrite the logarithm in terms of inverse trigonometric (or hyperbolic) functions. Not sure if that helps you do the integral, though. Also, did you try giving Mathematica some Assumptions as to the ranges of the variables? Sometimes if it knows a variable is real and positive, it is able to simplify, because now it knows what branch every function needs to be evaluated on.
 

Related to Solving Tough Integral with Log - Need Help

1. How do I approach solving a tough integral with log?

When solving an integral with log, it is important to first identify the type of log function present. If it is a basic log function, you can use the properties of logarithms to simplify the integral. If it is a natural log function, you can use the substitution method to simplify the integral.

2. What are the properties of logarithms that can help simplify an integral?

The main properties of logarithms that can be used to simplify an integral are the product, quotient, and power rules. These rules allow you to rewrite the integral in a simpler form, making it easier to solve.

3. How do I use the substitution method to solve an integral with natural log?

The substitution method involves substituting a variable for the expression inside the natural log function. This allows you to rewrite the integral in a simpler form, typically involving polynomials or trigonometric functions. Once the integral is in this form, you can use traditional integration techniques to solve it.

4. What are some common mistakes to avoid when solving a tough integral with log?

One common mistake is forgetting to use the properties of logarithms to simplify the integral. Another mistake is not properly substituting when using the substitution method. It is also important to carefully check the final answer, as it is easy to make computational errors when dealing with multiple steps.

5. Are there any tips for solving tough integrals with log more efficiently?

One tip is to practice identifying the type of log function present and being familiar with the properties of logarithms. Another tip is to check your answer by differentiating it to ensure it is correct. Additionally, breaking down the integral into smaller, manageable parts can make it easier to solve. And as always, practice and perseverance are key to becoming more efficient at solving tough integrals with log.

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