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

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Professor PC Thomas Classes · Answer

(a, c): Let $I=\int_{a}^{b} x f(x) d x$\ $I=\int_{a}^{b}(a+b-x) f(a+b-x) d x$\ $I=\int_{a}^{b}(a+b) f(a+b-x) d x$ $-\int_{a}^{b} x f(a+b-x) d x$\ $I=\int_{a}^{b}(a+b) f(x) d x-\int_{a}^{b} x f(x) d x$\ $	herefore \quad I=\frac{a+b}{2} \int_{a}^{b} f(x) d x$\ $=\frac{a+b}{2} \int_{a}^{b} f(a+b-x) d x$\