I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is-(adsbygoogle = window.adsbygoogle || []).push({});

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

i.e.g'(x) = f(x) ⇒ Indefinite Integral of f(x) is g(x)

In mathematical notation we write-∫f(x) dx = g(x) + Cif and only ifg'(x) = f(x) (as d/dx(g(x)+C) = f(x)

And the definition of definite integral is-

My question is that, firstly, in the definition of an indefinite integral, how can we be sure thatLetf(x)be continuous on[a,b]. IfG(x)is continuous on[a,b]andG'(x)=f(x)for allx (a,b), thenGis called an anti-derivative off.

We can construct anti-derivatives by integrating. The function

F(x)= [itex] \int_a^x f(t) dt [/itex]

is an anti-derivative forfsince it can be shown thatF(x)constructed in this way is continuous on[a,b]andF'(x) = f(x)for allx (a,b).

g(x)would have to be continuous∀ x ∈ R? And in the definition of a definite integral, how can we be sure thatg(x)would have to be continuous forx ∈ (a,b)?

Just becauseg'(x) = f(x), is this enough to say thatg(x)is a continuous function for∀ x ∈ Randx ∈ (a,b)respectively in the 2 definitions?g'(x)exists meansg(x)is a continuous function. But we don't know whetherg'(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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# Continuity in Integrals and Antiderivatives

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