Q80 / 120 · VITEEE 2016 · Mathematics

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

  • A
    $\frac{a+b}{2} \int_{a}^{b} f(b-x) d x$
  • B
    $\frac{a+b}{2} \int_{a}^{b} f(b+x) d x$
  • C
    $\frac{b-a}{2} \int_{a}^{b} f(x) d x$
  • Correct answer
    D
    $\frac{a+b}{2} \int_{a}^{b} f(x) d x$

Solution

Let $\mathrm{I}=\int_{a}^{b} x f(x) d x$ Let $\mathrm{a}+\mathrm{b}-\mathrm{x}=\mathrm{z} \Rightarrow-\mathrm{dx}=\mathrm{dz}$ When $x=a, z=b$ and when $x=b, z=a$ $$ \begin{array}{l} \therefore \quad \mathrm{I}=-\int_{b}^{a}(a+b-z) f(z) d z \\ \mathrm{I}=(a+b) \int_{b}^{a} f(x) d x-\int_{a}^{b} x f(x) d x \\ \mathrm{I}=(a+b) \int_{b_{b}}^{a} f(x) d x-\mathrm{I} \\ 2 \mathrm{I}=(a+b) \int_{a}^{b} f(x) d x \\ \text { Hence, } \mathrm{I}=\left(\frac{a+b}{2}\right) \int_{a}^{b} f(x) d x \end{array} $$

Hence, the correct answer is option: 'D'.

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