Integral Equality: \int f(x)g(x) = \int f(x) * \int g(x)

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In summary, integral equality is a mathematical concept that states the integral of a product of two functions is equal to the product of their integrals. This allows for simplification and solving of complex integration problems. It can be applied to all continuous and integrable functions, with some exceptions. It is closely related to the fundamental theorem of calculus, as the product of two functions can be thought of as the composition of their antiderivatives.
  • #1
Vadim
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just want to know if [tex]\int f(x)g(x)=\int f(x) * \int g(x)[/tex]
 
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  • #2
No, it is untrue.
 
  • #3
damn, that's what i thought, i just wanted to double check.
 
  • #4
Because (f(x)g(x))' is not f'(x)g'(x). It is (f(x)g(x))'= f'(x)g(x)+ f(x)g'(x). The "reverse" of that is "integration by parts" which can sometimes be used to integrate a product.
 

Related to Integral Equality: \int f(x)g(x) = \int f(x) * \int g(x)

1. What is integral equality?

Integral equality refers to the mathematical concept that states the integral of a product of two functions is equal to the product of their integrals. In other words, if we have two functions f(x) and g(x), the integral of their product, ∫f(x)g(x), is equal to the product of their individual integrals, ∫f(x) * ∫g(x).

2. What is the significance of integral equality?

Integral equality is significant because it allows us to simplify and solve complex integration problems. By breaking down a product of functions into the product of their individual integrals, we can use known integration techniques to solve the problem more easily.

3. Can integral equality be applied to all functions?

Yes, integral equality can be applied to all functions as long as they are continuous and integrable. This means that the function must be defined and have a finite integral over the interval of integration.

4. Are there any exceptions to integral equality?

Yes, there are some special cases where integral equality does not hold. For example, if the functions f(x) and g(x) are not continuous or if their product is not integrable over the interval of integration, then integral equality cannot be applied.

5. How is integral equality related to the fundamental theorem of calculus?

Integral equality is closely related to the fundamental theorem of calculus, which states that the integral of a function f(x) can be evaluated by finding its antiderivative F(x) and evaluating it at the upper and lower limits of integration. In the case of integral equality, we can think of the product of two functions as the composition of their antiderivatives, which is then evaluated at the limits of integration.

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