Definition:

`int_a^b f(x) dx = [ F(x) ]_a^b= F(b) - F(a)` is called the definite integral of `f(x)` between the limits `a` and `b`.
where `d /dx (F(x))= f(x)`

Note : The word limit here is quite different as used in differential calculus.

Important Points:
(I) If `int_a^b f(x) dx =0`, then the equation `f(x)=0` has at least one root in (a, b) provided `f` is continuous in `(a, b)`.

Note that the converse is not true.

(II) `lim_(n->oo) (int_a^b f_n (x) dx)= int_a^b (lim_(n->oo) f_n (x) )dx`

(III) `int_a^b f(x) *d (g (x) )= int_(g^(-1) (a))^ ( g^(-1) (b)) f(x) * g'(x) dx`.

(IV) If `f(x)` is continuous in `(a, b)`, Then `int_a^b d/dx (f (x)) = [F(x)]_a^b` and if `f(x)` is discontinuous in `(a, b)` at
`x =c in (a,b) ,` then `int_a^b d/dx (f (x))= [ f(x) ]_a^b + [f(x)]_(c^+)^b`

(V) lf `g(x)` is the inverse of `f (x)` and `f (x)` has domain `x in [a,b]` where `f(a) =c` and `f (b) = d` then the value of
`int_a^bf(x) dx + int_c^d g(y) dy = (bd -ac)`