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Q. If $\int\limits^{a}_{0} f\left(2a-x\right)dx = m$ and $ \int\limits^{a}_{0} f\left(x\right)dx = n , $ then $ \int\limits^{2a}_{0} f\left(x\right)dx $ is equal to

BITSATBITSAT 2018

Solution:

Put $x =2 a - t$
so that $dx =- dt$
when $x = a , t = a$ and when $x =2 a , t =0$
$\int\limits_{0}^{2} f(x) d x=\int\limits_{0}^{a} f(x) d x+\int\limits_{0}^{a} f(2 a-t) d t=n+m$