As we know, antiderivative or indefinite integral is the function the derivative of which gives the actual function. Let $F(x)$ be the derivative of $f(x)$ ie. the instantaneous rate of change of $f(x)$ with respect to $x$ is $F(x)$ . $$\dfrac{d{f(x)}}{dt} = F(x).$$ Now $$ d{f(x)} = F(x)\,dx,$$ right? Then writting indefinite integral on both side, we get, $$\int d{f(x)} = \int F(x)\,dx.$$ Then abrubtly many books do like that $$ f(x) = \int F(x)\,dx + C.$$ Really? What does this $\int$ mean? Summation, right? Then how can summation of $d{f(x)}$ give the function? It is the change and not the function. What about $C$? My physics book tells that it is the initial value ie. $v_0, a_0 , x_0$ . But it can be anything, right? So, how can this process give the function?

As we know, antiderivative or indefinite integral is the function the derivative of which gives the actual function. Let $F(x)$ be the derivative of $f(x)$ ie. the instantaneous rate of change of $f(x)$ with respect to $x$ is $F(x)$ . $$\dfrac{d{f(x)}}{dx} = F(x).$$ Now $$ d{f(x)} = F(x)\,dx,$$ right? Then writing indefinite integral on both side, we get, $$\int d{f(x)} = \int F(x)\,dx.$$ Then abruptly many books do like that $$ f(x) = \int F(x)\,dx + C.$$ Really? What does this $\int$ mean? Summation, right? Then how can summation of $d{f(x)}$ give the function? It is the change and not the function. What about $C$? My physics book tells that it is the initial value i.e. $v_0, a_0 , x_0$ . But it can be anything, right? So, how can this process give the function?

As we know, antiderivative or indefinite integral is the function the derivative of which gives the actual function. Let $F(x)$ be the derivative of $f(x)$ ie. the instantaneous rate of change of $f(x)$ with respect to $x$ is $F(x)$ . $$\dfrac{d{f(x)}}{dt} = F(x)$$ . Now $$ d{f(x)} = F(x).dx$$ , right?? Then writting indefinite integral on both side, we get, $$\int d{f(x)} = \int F(x).dx$$ . Then abrubtly many books do like that $$ f(x) = \int F(x).dx + C$$ . Really??? What does this $\int$ mean?? Summation, right?? Then how can summation of $d{f(x)}$ give the function?? It is the change and not the function. What about $C$ ?? My physics book tells that it is the initial value ie. $v_0, a_0 , x_0$ . But it can be anything, right?? So, how can this process give the function?? Please help.

As we know, antiderivative or indefinite integral is the function the derivative of which gives the actual function. Let $F(x)$ be the derivative of $f(x)$ ie. the instantaneous rate of change of $f(x)$ with respect to $x$ is $F(x)$ . $$\dfrac{d{f(x)}}{dt} = F(x).$$ Now $$ d{f(x)} = F(x)\,dx,$$ right? Then writting indefinite integral on both side, we get, $$\int d{f(x)} = \int F(x)\,dx.$$ Then abrubtly many books do like that $$ f(x) = \int F(x)\,dx + C.$$ Really? What does this $\int$ mean? Summation, right? Then how can summation of $d{f(x)}$ give the function? It is the change and not the function. What about $C$? My physics book tells that it is the initial value ie. $v_0, a_0 , x_0$ . But it can be anything, right? So, how can this process give the function?