Show that `int0af(x)g(x)dx=2int0af(x)dx` if f and g defined as `f(x)" "=" "f(a-x)` and `g(x)" "+g(a-x)=" "4`

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

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

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

int{f(x)g'(x)-f'g(x)}dx equals

Evaluate: int[f(x)g'(x)-f'(x)g(x)]dx

(A) For each of the following function f(x), verify that: int_0^2 f(x) dx=int_0^1 f(x) dx+int_1^2 f(x) dx: (i) f(x)=x+2 (ii) f(x)= x^2+2 (iii) f(x)= e^x (B) For each of the following pairs of function f(x) and g(x) , verify that: int_0^1 [f(x)+g(x)]dx=int_0^1 f(x) dx+int_0^1 g(x) dx: (i) f(x)=1,g(x) = x^2 (ii) f(x)=e^x, g(x)=1

If int f(x)dx=g(x) then int f^(-1)(x)dx is

Prove that int_0^a f(x)dx=int_0^af(a-x)dx , hence evaluate int_0^pi(x sin x)/(1+cos^2 x)dx

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