Indefinite integral of any function is not unique because by putting different values of constant in it, we can get different values of integral. The definite integral of an}? function has a unique value. If f(x) is any function, then its definite integral is denoted by \(\int ^b _a\) f(x) dx, where b is upper limit and a is lower limit. Hence, when we integrate any function within two limits, then it is called definite integral.
Let ∫f(x) dx = F(x) + C ...(1)
(Indefinite integral of the function) Now, putting upper limit b in place of the variable
∫f(x) dx = F(b) + C ...(2)
Now, putting lower limit a in place of the variable
∫f(x) dx = F{a) + C ...(3)
Substracting equation (3) from equation (2), we get
\(\int ^b _a\) f(x) dx = [F(b) + C) - [F(a) + C]
= m-m
The arbitrary constant disappers in evaluation of the value of the definite integral.
Definite integral can also be represented as :
\(\int ^b _a\) f(x) dx = [F(x)fa = F(b) - F(a)
[Here, F(x) is an anti-derivative : F(x) \(\frac{d}{dx}\) f(x) = f(x)]
Definite integral also be represented as the limit of a sum.
Let f be continuous function defined in the closed inverval [a, b] and F(x) be an antiderivative of f(x) then, definite integral of f(x) in closed interval [a, b] is denoted by \(\int ^b _a\) f(x) dx
and \(\int ^b _a\) f(x) dx = \([F(x)]^b_a\) = F(b) - F(a)
where a and b are the lower and upper limits of the integral. This definite integral has a unique value.
