Page 295 - DMTH401_REAL ANALYSIS

Unit 24: Fundamental Theorem of Calculus

Notes
0 x + h
ò [f(t) f(x )]dt £ - h . Therefore
-
0
0 x
F(x + h) F(x ) 
-
0 0 - f(x ) £ <  , if h <  .
h 0 2
-
F(x + h) F(x)
Therefore, lim 0 f(x ), i.e., F'(x ) = f(x )
0
0
0
h 0 h
which shows that F is differentiable at x and F'(x ) = f(x ). From Theorem 1, you can easily
0 0 0
deduce the following theorem:
Theorem 2: Let f: [a, b]  R be a continuous function. Let F : [a, b]  R be a function defined by
x
F(x) = ò f(t) dt, x E[a,b].
a
Then F'(x) = f(x), a £ x £ b.
This is the first result which links the concepts of integral and derivative. It says that, if f is
continuous on [a,b] then there is a function F on [a, b] such that F'(x) = f(x)," x [a,b].
Î
You have seen that if f: [a, b]  R is continuous, then there is a function F: [a, b]  R such that F'
(x) = f(x) on [a, b]. Is such a function F unique? Clearly the answer is 'no'. For, if you add a
x
c
constant to the function F, the derivative is not altered. In other words, if G(x) = + ò f(t) dt for
1
a £ x £ b then also G' (x) = f(x) on [a, b].
Such a function F or G is called primitive off. We have the formal definition as follows:

24.2 Primitive of a Function

If f and F are functions defined on [a,b] such that F’(x) = f(x) for x Î [a,b] then F is called a
'primitive' or an 'antiderivative' of f on [a,b].
Thus from Theorem 1, you can see that every continuous function on [a,b] has a primitive. Also
there are infinitely many primitives, in the sense that adding a constant to a primitive gives
another primitive.

"Is it true that any two primitives differ by a constant?"
The answer to this question is yes. Indeed if F and G are two primitives of f in [a,b], then
=
F'(x) G'(x) = f(x) " x [a,b] and therefore [F(x) – G(x)’ = 0. Thus F(x) – G(x) = k (constant), for x
Î
Î [a,b].
Let us consider an example.

Example: What is the primitive of f(x) = log x in [1, 2]
d
-
Solution: Since (xlog x x) = log x " x [1,2], therefore F (x) = x log x – x is a primitive of f in
Î
dx
[1, 2].
Also G(x) = x lag x – x + k, k being a constant, is a primitive of f.

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