JEE Main2003MathematicsDefinite IntegrationActual

If f ( a + b - x )= f ( x ) then _a^b xf ( x ) dx is equal to

Options

Aa+b 2 _a^b f(a+b-x) d x
Ba+b 2 _a^b f(b-x) d x
Ca+b 2 _a^b f(x) d x
Db-a 2 _a^b f(x) d x

Correct answer

A. a+b 2 _a^b f(a+b-x) d x

Step-by-step solution

I= _a^b x f(x) d x= _a^b(a+b-x) f(a+b-x) d x=(a+b) _a^b f(a+b-x) d x- _a^b x f(a+b-x) d x=(a+b) _a^b f(a+b-x) d x- _a^b x f(x) d x2 I=(a+b) _a^b f(x) d xI= (a+b) 2 _a^b f(x) d x ; I= (a+b) 2 _a^b f(a+b-x) d x

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