Prove that `int_0^t f(x)g(t-x)dx=int_0^t g(x)f(t-x)dx`

If f and g are continuous functions on [0,a] such that f(x)=f(a-x) and g(x)+g(a-x)=2 then show that int_(0)^(a)f(x)g(x)dx=int_(0)^(a)f(x)dx

If fandg are continuous function on [0,a] satisfying f(x)=f(a-x)andg(x)+g(a-x)=2 then show that int_(0)^(a)f(x)g(x)dx=int_(0)^(a)f(x)dx

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

Let f(x) be a continuous function AA x in R, except at x=0, such that int_(0)^(a)f(x)dxa in R^(+) exists.If g(x)=int_(x)^(a)(f(t))/(t)dt, prove that int_(0)^(a)f(x)dx=int_(0)^(a)g(x)dx

If f\ a n d\ g are continuious on [0,\ a] and satisfy f(x)=f(a-x)a n d\ g(x)+g(a-x)=2. show that int_0^af(x)g(x)dx=int_0^af(x)dx

int{f(x)+-g(x)}dx=int f(x)dx+-int g(x)dx

Property 12:int_(a)^(a+nT)f(x)dx=n int_(0)^(T)f(x)dx where T is period of f(x)

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

If f(x) and g(x) be continuous functions in [0,a] such that f(x)=f(a-x),g(x)+g(a-x)=2 and int_0^a f(x)dx=k , then int_0^a f(x)g(x)dx= (A) 0 (B) k (C) 2k (D) none of these

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