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Basic Mathematics-II

Notes as the Fundamental Theorem of Calculus, between indefinite integral and definite integral
which makes the definite integral as a practical tool for science and engineering.

1.1 Integration as an Inverse Process of Differentiation

Integration is defined as the inverse process of differentiation. Rather than differentiating a
function, we are provided the derivative of a function and requested to locate its primitive, i.e.,
the original function. This type of process is known as Integration or anti differentiation.

1.1.1 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus identifies the relationship among the processes of
differentiation and integration. That relationship says that differentiation and integration are
inverse processes.
The Fundamental Theorem of Calculus: Part 1

If f is a continuous function on [a,b], then the function indicated by
x
g ( )   f ( )du for   ,a b 
x
x
u
a
is continuous on [a,b], differentiable on (a,b) and g’(x) = f(x).

If f(t) is continuous on [a,b], the function g(x) which is equal to the area enclosed by the u-axis and
the function f(u) and the lines u=a and u=x will be continuous on [a,b] and differentiable on (a,b).
Most prominently, while we differentiate the function g(x), we will discover that it is equal to
f(x). The above graph demonstrates the function f(u) and the area g(x).

The Fundamental Theorem of Calculus : Part 2

If f is a continuous function on [a,b], then
b
x
a
F
b
 f ( )dx  F ( )– ( )
a
where F is any antiderivative of f.
If f is continuous on [a,b], the definite integral with integrand f(x) and limits a and b is just equal
to the value of the antiderivative F(x) at b minus the value of F at a. This property permits us to
simply resolve definite integrals, if we can locate the antiderivative function of the integrand.
Part one and part two of the Fundamental Theorem of Calculus can be combined as below.
Combining the Fundamental Theorem of Calculus Part 1 and Part 2

Let f be a continuous function on [a,b].

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