`(A)text(Properties )`:

P-1: `int_a^b f(x) dx =int_a^b f(t) dt`;

P-2: `int_a^bf(x) dx =-int_b^a f(x) d(x)`

P-3: `int_a^b f(x) = int_a^c f(x) +int_c^b f(x) dx` provided `f` has a piece wise continuity or when `f` is not uniformly defined in `(a, b)`

Integral is broken at points of discontinuity or at the points where definition of `f` changes.

P-4: `int_-a^a f(x) dx =int _0^a (f(x) +f(-x))dx = [tt( (0,text(if) f(x)quad is quadodd),( 2 int_0^a f(x) dx, text(if) f(x)quad isquad even) )`


`text( Proof: )quad` `I= int_-a^a f(x) dx= int_-a^0 f(x)dx + int_0^a f(x) dx ` Put `x = - t` in first integral.

`quadquadquadquadquadquadquad= int_a^0f(-t) (-dt) + int_0^a f(x) dx =int_0^a (-t) dt + int_0^a f(x) dx=int_0^a f(-x) dx +int_0^a f(x)dx`

`quadquadquadquadquadquadquadquadquadquadquad=int_0^a { f(x) +f (-x)}dx`

P-5: `int_a^b f(x) dx = int_a^b f(a+b-x) dx ` or `int+0^a f(x) dx =int_0^a f (a-x)dx`

`text( Proof:)quad` `I =int_a^b (a+b-x)dx` put `a+b-x =t` `=> -dx=dt` & `I= int_b^a f(t) (-dt)`

`quadquadquadquadquadquadquad= int_a^b f(t) dt = int_a^bf(x) dx`

P-6: `int_0^2a f(x) dx =int_0^a f(x) dx+int_0^a f(2a-x) dx => [tt( (0,text(if) f(2a-x) =-f(x)), (2 int_0^a f(x) dx , text(if) f (2a-x) =f(x)))`

`text( Proof:)quad` `I= int_0^2a f(x) dx =int_0^a f(x) dx + int_a^2a f(x)dx ` put `x=2a-t` in2nd integral

`quadquadquadquadquadquadquadquadquad=> I= int_0^a f(x) dx +int_a^0 f(2a-t)(-dt) = int_0^a f(x) dx + int_0^af (2a-x)dx`

`quadquadquadquadquadquadquad= int_0^a { f(x) + f (2a-x)}dx`

P-7 : `int_0^(nT)f(x) dx =n int_0^Tf(x) dx` where `f(T+x)=f(x) n in I`