Differentiation of integrals and integration(?)

In summary, the formula provided is known as the Leibniz integral rule and can be used to evaluate integrals of certain types. These types include those with a derivative in the integrand and those with varying limits of integration. The procedure involves taking the derivative of the upper limit, multiplying by the value of the function at the upper limit, taking the derivative of the lower limit, multiplying by the value of the function at the lower limit, and then integrating the partial derivative of the function with respect to x over the original limits.
  • #1
ShayanJ
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I heard that the formula below can be used to evaluate some kinds of integrals but I can't find what kinds and how to do it.Could someone name those kinds and also the procedure?

[itex]
\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) b'(x) - f(x,a(x)) a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) dt
[/itex]


Thanks
 
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  • #3
That doesn't contain what I wanted.
I have seen another page from wikipedia too.That one contains some integration using the formula but I can't understand the general procedure.
 

Related to Differentiation of integrals and integration(?)

1. What is the difference between differentiation and integration?

Differentiation and integration are two fundamental operations in calculus. Differentiation is the process of finding the rate of change of a function, while integration is the opposite process of finding the total accumulation of a function. In simpler terms, differentiation is like finding the speed of a moving object, while integration is like finding the distance traveled.

2. Why is the derivative of an integral equal to the original function?

This is known as the Fundamental Theorem of Calculus. It states that the derivative of an integral is equal to the original function because integration is the reverse operation of differentiation. In other words, finding the derivative of an integral is like "undoing" the integration process.

3. What is the process for finding the derivative of an integral?

The process for finding the derivative of an integral is known as the Second Fundamental Theorem of Calculus. It involves using the chain rule to differentiate the inside function, and then multiplying by the derivative of the inside function. This can be expressed as d/dx ∫f(x)dx = f(x).

4. Can integration and differentiation be used to solve real-world problems?

Yes, integration and differentiation are essential tools in physics, engineering, economics, and many other fields. They can be used to model and solve real-world problems involving rates of change and accumulation.

5. Are there any special techniques for solving integrals?

Yes, there are several techniques for solving integrals, such as substitution, integration by parts, and trigonometric substitution. These techniques can make solving integrals easier and more efficient, especially for more complex functions.

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