Property 12: `int_a ^(a+nT) f(x) dx = n int_0 ^T f(x) dx` where T is period of f(x)

Property 11: int_0 ^(nT) f(x)= n int_0 ^T f(x) dx where T is the period of the function and n is integer

Property 13:int_(mT)^(nT)f(x)dx=(n-m)int_(0)^(T)f(x)dx where T is period of f(x) and m and n are integer

int_0^a f(a-x) dx=

Property 14:int_(a+nT)^(b+nT)f(x)dx=int_(a)^(b)f(x)dx where T is the period of the f(x) and n is integer

Statement 1: int_(0)^(10)(2^(x))/(2^([x]))dx=(10)/(ln2) (where [.] denotes the greatest integer function) because Statement 2: int_(0)^(nT)f(x)dx=n int_(0)^(T)f(x)dx if T is the fundamental period of f(x), n is rational number.

int _(0) ^(2a) f (x) dx = int _(0) ^(a) f (x) dx + int _(0)^(a) f (2a -x ) dx

Statement-1: int_(0)^(npi+v)|sin x|dx=2n+1-cos v where n in N and 0 le v lt pi . Stetement-2: If f(x) is a periodic function with period T, then (i) int_(0)^(nT) f(x)dx=n int_(0)^(T) f(x)dx , where n in N and (ii) int_(nT)^(nt+a) f(x)dx=int_(0)^(a) f(x) dx , where n in N

Prove that : int_(-a)^(a)f(x)dx =2 int_(a)^(0) f(x)dx, if f(x) is even funtion =0 , if f(x) is off fuction.

Property 3:int_(a)^(b)f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx

Statement I int_0 ^ (npi + t) | sinx | dx = (2n + 1) -cost, (0 leq t leq pi) Statement II int_a ^ bf (x) dx = int_a ^ ef (x) dx + int_e ^ bf (x) dxint_0 ^ (na) f (x) dx = nint_0 ^ af (x) dx "if" f (a + x) = f (x)