Prove that `int_a^bf(x)dx=int_(a+c)^(b+c)f(x-c)dx` , and when `f(x)` is odd function, `int_(-a)^af(x)dx=0`

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

If int_(a)^(b)f(x)dx=int_(a)^(b)phi(x)dx , then-

int_(0)^(na)f(x)dx=nint_(0)^(a)f(x)dx if-

int_(a)^(b)f(x)dx is equal to-

Prove that int_(0)^(2a)f(x)dx=int_(0)^(a)[f(a-x)+f(a+x)]dx

Show : int_(a)^(b)f(kx)dx=(1)/(k)int_(ka)^(kb)f(x)dx

If f(x) is an odd function of x, then show that, int_(-a)^(a)f(x)dx=0 .

Which of the following is incorrect? int_(a+ c)^(b+c)f(x)dx=int_a^bf(x+c)dx int_(ac)^(b c)f(x)dx=cint_a^bf(c x)dx int_(-a)^af(x)dx=1/2int_(-a)^a(f(x)+f(-x)dx None of these

Show that int_(0)^(a)f(x)g(x)dx=2int_(0)^(a)f(x)dx , if f and g are defined as f (x) = f (a - x) and g(x) + g(a - x) = 4

if int_0^1 f(x)dx=1,int_0^1 xf(x)dx=a and int_0^1 x^2f(x)dx=a^2 , then int_0^1(a-x)^2f(x)dx is equal to