# Differentiation under the integral sign

• Wolfxphysics12
In summary, the conversation discusses the concept of introducing a parameter into an integral, which is similar to multiplying by 1 in algebra. By defining a new integral with the parameter, and differentiating it, it becomes easier to solve the original integral. Many examples and resources are available online for further understanding.
Wolfxphysics12
I have read about this method , and how feynman utilized this method. I like doing integrals for fun, but I can't seem to understand the conceptual idea on how to introduce a parameter into the integral. Can someone , in detail, explain to me how to introduce the parameter into the integral ? (Thank you :) )

Conceptually it is the same as multiplying a number by 1 in regular algebra: you can multiply the integrand by 1 without changing the integrand. If you are clever, it will make the integral easier to do.

i.e. we want to investigate $$I(n)=\int_0^\infty x^ne^{-x}\; dx$$

Start out by defining:
$$I(n,\lambda)=\int_0^\infty x^ne^{-\lambda x}\; dx$$ ... bearing in mind that ##I(n,\lambda)=I(n)## if ##\lambda = 1##

Now differentiate both sides by lambda.

Why may we suspect that this would help?
Because we can do I(n=0), so we only need a way to get to I(n-1) from I(n), which we can repeat until n=0 and the answer falls out, and we know what differentiating an exponential function does. This is actually what you'd do by using differentiation by parts.

There are many examples online

## 1. What is differentiation under the integral sign?

Differentiation under the integral sign is a method used to find the derivative of a function that is defined by an integral. It allows us to take the derivative of a function with respect to a variable that appears as one of the limits of integration in the integral.

## 2. Why is differentiation under the integral sign useful?

This method is useful because it simplifies the process of finding the derivative of a function defined by an integral. It also allows us to evaluate integrals that would otherwise be difficult or impossible to solve.

## 3. How does differentiation under the integral sign work?

The basic idea is to treat the variable of integration as a constant and differentiate the integrand with respect to the other variable. Then, we integrate the resulting function with respect to the variable of integration, and this gives us the derivative of the original function.

## 4. What are the conditions for using differentiation under the integral sign?

In order to use this method, the integrand must be a continuous function of two variables, the limits of integration must be constants, and the partial derivative of the integrand with respect to the variable of integration must exist and be continuous.

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

Yes, this method may not work for all functions. It is important to check the conditions mentioned above before using this method. Additionally, the resulting derivative may be an implicit function and may need further manipulation to be expressed explicitly.

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