If f(a+b-x)=f(x), then int_a^bxf(x)dx is equal to?

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1 Answer
Sep 30, 2017

Answer is :

(a+b)/2 int_a^b f(x) dx

Explanation:

Let "I" be the integral,

I = int_a^b xf(x) dx .......... (1)

Using the property of definite integrals,

I = int_a^b (a+b-x) f(a+b-x) dx

It is given that, f (a+b-x) = f(x)

implies I = int_a^b (a+b-x) f(x) dx

implies I = (a+b) int_a^bf(x) dx - int_a^b x f(x) dx .......... (2)

Adding (1) and (2),

2I = (a+b) int_a^b f(x) dx

implies I = (a+b)/2 int_a^b f(x) dx