It doesn't matter whether the left side gives a constant or not. The right side does give a constant that is not generally the same as the one on the left side. However, it can be shown that only one constant is needed.brushman said:Oh, so the indefinite integral of 0 gives you a constant. So I'm still confused with if the left side simplifies to f(x), or f(x) + a constant.
Besides the fact that I improperly integrated zero, I don't see any errors on the left side of my evaluations. Perhaps they are both correct, but method two gives you more information about the original function f?
vela said:Neither looks right to me. When you start with
$$\frac{d}{dx} f(x) = 0,$$ integrating both sides gives you
$$\int \frac{d}{dx} f(x)\,dx = \int 0\,dx.$$ The righthand side isn't necessarily equal to 0.
This is an indefinite integral (i.e. it's an anti-derivative), it's not a definite integral, so the Riemann sum comment doesn't apply here.WWGD said:Doesn't this violate linearity of the integral? If we have ∫0dx , then use 0=0.0 , so
∫0dx =0[∫0dx] =0
Moreover: If we used the perspective of Riemann sums, then ∫0dx is the area under
a curve with height zero. And ∫0dx=∫(c-c)dx=∫cdx-∫cdx.
The term "integrate" in this context refers to finding the antiderivative of a given function. In simpler terms, it is the reverse process of differentiation, where we are trying to find the original function from its derivative.
The integral notation for integrating a function is written as ∫ f(x) dx. It is read as "the integral of f(x) with respect to x". The integral sign (∫) represents the process of integration, while the function (f(x)) and the dx represent the function and the variable with respect to which we are integrating, respectively.
The notation "d/dx" represents the derivative operator, which is used to denote the derivative of a function with respect to the variable x. In the context of integration, it indicates that we are finding the antiderivative of the function f(x) with respect to x.
The process for integrating a function with respect to x involves finding the antiderivative of the given function using the rules of integration. These rules include the power rule, product rule, quotient rule, and chain rule. Once the antiderivative is obtained, the constant of integration is added to the solution to account for all possible solutions to the original function.
No, not all functions can be integrated with respect to x. Some functions, such as trigonometric functions and logarithmic functions, have specific rules for integration that need to be applied. However, there are some functions that are not possible to integrate using these rules, and they are known as non-integrable functions.