How does differentiation under the integral sign

In summary, the conversation discusses the method of integration that involves adding parameters to functions. The individual is trying to understand how to use this technique and when it is applicable. They reference resources such as Woods' 'Advanced Calculus' and a pdf on the topic, and also mention Feynman's use of the method in his work. They question whether all integrals that can be solved using this technique involve a parameter and a variable.
  • #1
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I've read about it before and now I'm trying to learn it myself from Woods 'Advanced Calculus' (as well as other resources like http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf)

In the pdf, it says the method concerns integrals that depend on a parameter...now couldn't we make any function depend on a parameter?

For example inserting a parameter into [tex]f(x)=x^2[/tex] so it becomes [tex]f(x)=\alpha x^2[/tex]

All the examples I've come across already have parameters in them so I'm not really sure. In 'Surely You're Joking, Mr Feynman!', Feynman says that he used this technique over and over again and often solved integrals that others couldn't using it. Would these integrals all be ones that depend on some parameter as well as a variable?

I'd like to know how exactly to use this method of integration and when to use it.
 
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  • #2
You can always add parameters to your functions. It is not useful everywhere, but you can do it.
 

FAQ: How does differentiation under the integral sign

1. How does differentiation under the integral sign work?

Differentiation under the integral sign is a mathematical technique used to find the derivative of a function that is written as an integral. It involves applying the chain rule of differentiation to the integrand and then integrating the result with respect to the variable of integration.

2. When is differentiation under the integral sign used?

This technique is used when the function to be differentiated cannot be easily expressed as a single algebraic expression. It is commonly used in physics and engineering to solve problems involving complex integrals.

3. What is the benefit of using differentiation under the integral sign?

By using this technique, we can avoid the tedious and time-consuming process of evaluating the integral first and then taking the derivative. It also allows us to solve difficult integrals that may not have a closed-form solution.

4. Are there any limitations to differentiation under the integral sign?

Yes, there are certain conditions that must be met for this technique to work. The integrand must be continuous and the bounds of integration must be constant. It also works best for single-variable functions, although it can be extended to multidimensional integrals.

5. Can differentiation under the integral sign be used for all types of integrals?

No, this technique is most effective for definite integrals, where the bounds of integration are known. It is not as useful for indefinite integrals, where the constant of integration would need to be determined separately.

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