If $f(a+b-x)=f(x)$, then $\int_\limits{a}^{b} x f(x) d x=$

$\frac{a+b}{2} \int_\limits{a}^{b} f(a+b+x) d x$

$\frac{a+b}{2} \int_\limits{a}^{b} f(b-x)$

$\frac{b-a}{2} \int_\limits{a}^{b} f(x) d x$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Application +4 | -1

f(a+b-x)=f(x)

, then

\int\limits_{{a}}^{b} x f(x) d x=

View answer and explanation

Correct option: C

\frac{a+b}{2} \int\limits_{{a}}^{b} f(x) d x

Option C (

\frac{a+b}{2} \int\limits_{{a}}^{b} f(x) d x

) is correct because using the property

f(a+b-x) = f(x)

simplifies the integral to this form.

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