Solve the following equations:


If $f(\mathrm{a}+\mathrm{b}+1-\mathrm{x})=f(\mathrm{x})$, for all $\mathrm{x}$, where $\mathrm{a}$ and $b$ are fixed positive real numbers, then

$\frac{1}{\mathrm{a}+\mathrm{b}} \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x}(f(\mathrm{x})+f(\mathrm{x}+1)) \mathrm{dx}$ is equal to :

  1. $\int_{a+1}^{b+1} f(x) d x$

  2. $\int_{a+1}^{b+1} f(x+1) d x$

  3. $\int_{\mathrm{a}-1}^{\mathrm{b}-1} f(\mathrm{x}+1) \mathrm{dx}$

  4. $\int_{\mathrm{a}-1}^{\mathrm{b}-1} f(\mathrm{x}) \mathrm{dx}$

Correct Option: , 4


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