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I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is-

And the definition of definite integral is-

My question is that, firstly, in the definition of an indefinite integral, how can we be sure that

Just because

*The indefinite integral of a function x with respect to f(x) is another function g(x) whose derivative is f(x).*

i.e.

In mathematical notation we write-

i.e.

**g'(x) = f(x) ⇒ Indefinite Integral of f(x) is g(x)**In mathematical notation we write-

**∫f(x) dx = g(x) + C**if and only if**g'(x) = f(x) (as d/dx(g(x)+C) = f(x)**And the definition of definite integral is-

*Let*

We can construct anti-derivatives by integrating. The function

**f(x)**be continuous on**[a,b]**. If**G(x)**is continuous on**[a,b]**and**G'(x)=f(x)**for all**x
(a,b)**, then**G**is called an anti-derivative of**f**.We can construct anti-derivatives by integrating. The function

**F(x)**= [itex] \int_a^x f(t) dt [/itex]*is an anti-derivative for*

**f**since it can be shown that**F(x)**constructed in this way is continuous on**[a,b]**and**F'(x) = f(x)**for all**x
(a,b)**.**g(x)**would have to be continuous**∀ x ∈ R**? And in the definition of a definite integral, how can we be sure that**g(x)**would have to be continuous for**x ∈ (a,b)**?Just because

**g'(x) = f(x)**, is this enough to say that**g(x)**is a continuous function for**∀ x ∈ R**and**x ∈ (a,b)**respectively in the 2 definitions?**g'(x)**exists means**g(x)**is a continuous function. But we don't know whether**g'(x)**exists**∀ x ∈ R**. Similarly, [itex] \int_a^x f(t) dt [/itex] is integral in the interval**(a,b)**. So how can we be certain that its anti-derivative would also be valid for only the interval**(a,b)**? Is this the definition of the indefinite integral that we can't question as to why would the -derivative be valid for only the interval**(a,b)**?

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