Property 1: Integration is independent of the change of variable. `int_a ^b f(x) dx = int_a ^b f(t) dt`

Prove that int_(a)^(b) f(x) dx= int_(a)^(b) f(a+b-x) dx

int_a^b[d/dx(f(x))]dx

Prove that the equality int_(a)^(b) f(x) dx = int_(a)^(b) f(a + b - x) dx

int _(a) ^(b) f (x) dx = int _(a) ^(b) f (a + b - x) dx

Property 4: If f(x) is a comtinuous function on [a;b] then int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx

Let f be a continous function on [a,b] . Prove that there exists a number x in [a, b] such that int _(a) ^(x) f(t) dt = int_(x) ^(b) f(t) dt .

If f(a+b-x0=f(x) , then prove that int_a^b xf(x)dx=((a+b)/2)int_a^bf(x)dxdot

Property 2: If the limits of a definite integral are interchanged then its value changes.int_(a)^(b)f(x)dx=-int_(b)^(a)f(x)dx

If f(a+b-x)=f(x),\ then prove that \ int_a^b xf(x)dx=(a+b)/2int_a^bf(x)dx