While analyzing the foundation of calculus I am finding that the notion of an integral is a special form of summation of differentials, and an indefinite integral is also an integral with limits $0$ to $x$ which is conventionally written without the limits,
$$ \bbox[color:#CE5C5C;]{ \sum_{0}^{x} f(x) \, \mathrm{d}x = \int_{0}^{x} f(x) \, \mathrm{d}x } = \int f(x) \, \mathrm{d}x $$
Please confirm if my assumption is correct, thanks.
... the notion of an integral is a special form of summation of differentials,
This is a great intuition for grasping the idea of integrals. Definite integrals are really designed to capture that notion of adding up tiny, infinitesimal quantities. And honestly, this idea alone is probably enough if you're doing physics or working on other practical applications.
But in mathematics, this simple intuition alone doesn't quite cut it when it comes to the level of rigor they need. This is mainly because the real number system cannot host any non-trivial infinitesimals other than $0$, and it is surprisingly hard to rigorize all the desiderata for infinitesimals.
So mathematicians had to come up with a more precise definition of integrals that avoids catastrophic consequences from any naïve notion of infinitesimals. One successful approach is to define a definite integral as the limit of Riemann sums. I'm guessing you're probably already familiar with this.
(It's worth noting that there exists a rigorous approach that actually concretizes the idea of regarding integrals as sums of infinitesimals. This approach, however, requires certain level of understanding of hyperreal number systems and non-standard analysis, which is somewhat more advanced than calculus.)
... and an indefinite integral is also an integral with limits to which is conventionally written without the limits
Well, that's not exactly right. The indefinite integral is actually the inverse operation of differentiation. That's why people often call it an antiderivative. And if you think about it, this definition doesn't really connect indefinite integrals to definite integrals directly.
That's what makes it so astonishing and marvelous that these two concepts are actually closely tied together. It's all thanks to that famous result we call the Fundamental Theorem of Calculus.
There are two kinds of integrals: indefinite and definite.
The indefinite integral of a function $f(x)$ is represented as:
$$ \int f(x) \, dx $$
The indefinite integral is an antiderivative, meaning a function $F(x)$ such that $F'(x) = f(x)$. A function can have multiple antiderivatives, resulting in a set of antiderivatives. If the domain of the function is an interval, then the antiderivatives of the function would be $F(x) + C$, where $F(x)$ is an antiderivative and $C$ is a constant. It is written as:
$$ \int f(x) \ dx = F(x) + C $$
The definite integral of a function $f(x)$ over the interval $[a, b]$ is represented as:
$$ \int_{a}^{b} f(x) \ dx $$
The definite integral is related to the area under the curve of $f(x)$ from $x = a$ to $x = b$. It can be defined in several ways, with the most basic being the Riemann integral definition. The definite integral can be thought of as the sum of infinitesimal differences over the interval $[a, b]$.
These two definitions are equivalent in a certain way due to the Fundamental Theorems of Calculus.
If $f$ is continuous on $[a, b]$ and $F(x)$ is defined as :
$$ F(x) = \int_{a}^{x} f(t) \, dt $$
Then $F$ is differentiable on $(a, b)$, and $F'(x) = f(x)$.
So, the indefinite integral can be viewed as a definite integral with a variable upper limit and an added constant $C$:
$$ \int f(x) \, dx = \int_{a}^{x} f(t) \, dt + C $$
It is a very bad habit to use the same name for the variable o be integrated and the limit. So better write$\int_0^x f(t)dt$. And your summation sign makes no sense; there should be a limit. $\int_0^x f(t)dt=F(x)-F(0)$ while $\int f(x)dx=F(x)+C$ where F is the antiderivative and C any constant.
