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

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Answer 1 · 2017-09-30T08:53:14

Answer is :

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

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 #