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Prove that: inta^b(f(x))/(f(x)+f(a+b-x...

Prove that: `int_a^b(f(x))/(f(x)+f(a+b-x))dx=(b-a)/2`

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True or False: underseta oversetbint (f(x))/(f(x)+f(a+b-x))dx=(a+b)/2

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If g(x)=(f(x))/((x-a)(x-b)(x-c)), where f(x) is a polynomial of degree lt3, then prove that (dg(x))/(dx)=|{:(1,a,f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)),(1,c,f(c)(x-c)^(-2)):} |divide|{:(a^(2),a,1),(b^(2),b,1),(c^(2),c,1):}|.

Let f be a one-one function such that f(x).f(y) + 2 = f(x) + f(y) + f(xy), AA x, y in R - {0} and f(0) = 1, f'(1) = 2 . Prove that 3(int f(x) dx) - x(f(x) + 2) is constant.

If f and g are continuous functions on [0,a] satisfying f(x)=(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 .