Differentiating under the integral

Thread starter ddcamp
Start date Oct 23, 2010
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Differentiating Integral

In summary, "Differentiating under the integral" is a mathematical technique used to find the derivative of a function defined by an integral. It is useful in situations where the function is dependent on a parameter or too complex to integrate directly. The steps include identifying the variable and limits of integration, differentiating both, and integrating the resulting expression. There are two special cases, and it has real-life applications in physics, engineering, and economics.

Oct 23, 2010

ddcamp

Homework Statement

integral 0 to infinite (x^3)(e^-8)(cos(5x)) dx

Homework Equations

integral by part?

The Attempt at a Solution

should I use integral by part?

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Oct 23, 2010

fzero

Science Advisor

Homework Helper

Gold Member

Do you mean

[tex] \int_0^\infty x^3 e^{-8x} \cos(5x) dx?[/tex]

You might want to use the identity

[tex] e^{iu} = \cos u + i \sin u[/tex].

and then integrate by parts.

1. What is "Differentiating under the integral"?

"Differentiating under the integral" is a mathematical technique that allows us to find the derivative of a function that is defined by an integral. It involves differentiating both the limits of integration and the integrand with respect to a variable, and then integrating the resulting expression.

2. When is "Differentiating under the integral" useful?

"Differentiating under the integral" is useful in situations where the function being integrated is dependent on a parameter, and we need to find the derivative of the integral with respect to that parameter. It is also useful when the integrand is too complex to integrate directly.

3. What are the steps to "Differentiating under the integral"?

The steps to "Differentiating under the integral" are as follows:1. Identify the variable of differentiation and the limits of integration2. Differentiate the limits of integration with respect to the variable3. Differentiate the integrand with respect to the variable4. Plug in the differentiated limits and integrand into the integral5. Integrate the resulting expression to find the derivative

4. Are there any special cases when using "Differentiating under the integral"?

Yes, there are two special cases when using "Differentiating under the integral":1. If the integrand is independent of the variable of differentiation, then the derivative of the integral is simply the integrand multiplied by the derivative of the limits.2. If the limits of integration are constants, then the derivative of the integral is equal to 0.

5. What are some real-life applications of "Differentiating under the integral"?

"Differentiating under the integral" has various applications in physics, engineering, and economics. It can be used to find the rate of change of a quantity, the slope of a curve, or the optimization of a system. Some examples include calculating the velocity of a moving object, finding the maximum profit in a business model, and solving differential equations.