Let F(x) be any antiderivative of f(x), then for any two values of the independent variable x, say a and b, the difference F (b) -F (a) is called the definite integral of f(x) from a to b and is denoted by f(x) dx. Thus

dx = F (b) - F (a)

where F(x) is any antiderivative of f(x). The numbers a and b appearing with the integral sing f in Equation are called the limits of integration; a is the lower limit and b is the upper limit. Usually, F (b) – F (a) is abbreviated by writing F (x) .

The reason for using the term “definite integral” follows from the fact that the value of the definite integral is independent of the particular choice of the antiderivative of f(x). For, if F (x) + c is any other antiderivative of f(x), then

dx = F(x) + c = [F (b) + c] - [ F(a) + c] = F(b) - F (a),

which is same as before.

Thus to find dx, it suffices to find an antiderivative of f(x), say F (x) and subtract the value of F (x) at the lower limit a from its value at the upper limit b.

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