Integral f (x) * g (x) doesn't equal
Integral f (x)dx * integral g (x)dx
Why?
the chain rule says that
d/dx(fg) = f'g + fg'
not f'*g'
The statement "Integral f (x) * g (x) doesn't equal Integral f (x)dx * Integral g (x)dx" is correct. This is because the product of the integrals of two functions is generally not equal to the integral of their product.
To understand why, let's consider the properties of integration. Integration is a way to find the area under a curve. For a single function, let's say f(x), the integral of f(x)dx over a given interval represents the area under the curve of f(x) within that interval.
When we have two functions, f(x) and g(x), and we want to find the integral of their product, the integral of f(x) * g(x)dx, it is not valid to simply multiply the integrals of f(x) and g(x) separately. This is because integration is not a linear operator, meaning that integrating the product of two functions cannot be broken down into the product of their separate integrals.
In other words, when we integrate the product f(x) * g(x)dx, we need to consider the interaction between the two functions. The integral of f(x) * g(x)dx takes into account how f(x) and g(x) vary together, not independently.
To properly evaluate the integral of the product, we usually have to use integration techniques such as substitution or integration by parts. These techniques help us handle the interaction between the two functions and find the appropriate result.