Question

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State first fundamental theorem of integral calculus.
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Answer 1

Answer: The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration.

Step-by-step explanation:

Answer 2

Answer:

The fundamental theorem of calculus connects differentiation and integration , and usually consists of two related parts . It can be used to find definite integrals without using limits of sums .

I will state the definitions of the first and of the second fundamental theorem of calculus as given by Wikipedia ( and by many calculus textbooks).

First fundamental theorem:

The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.[...]

Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by

f(x)= integral f(t)dt. from a to x

Then, F is uniformly continuous on [a, b], differentiable on the open interval (a, b), and

F`(x) = f(x)

for all x in (a, b).

So in the definition above F is an antiderivative of f on [a,b] .

Second fundamental theorem , or Newton-Liebniz axiom .This part of the theorem depicts how to evaluate definite integrals by finding and evaluating an antiderivative at the upper and lower limits of integration :

Let f and F be real-valued functions defined on a closed interval [a, b] such that the derivative of F is f. That is, f and F are functions such that for all x in [a, b],

F'(x) =f(x)

If f is Riemann integrable on [a, b] then

integral of f(x)dx from a to b = F(b)-F(a)

When an antiderivative F exists, then there are infinitely many antiderivatives for f, obtained by adding to F an arbitrary constant. Also, by the first part of the theorem, antiderivatives of f always exist when f is continuous.